Caveat--I address this topic as an Evangelical Christian writer who has included aliens in science fiction stories and feel justified in doing so. The objections (and answers) below were written by me, but based on things I've encountered elsewhere:
O1. No aliens (or alien planets) are mentioned anywhere in the Bible, so there must not be any.
A1. A counter-argument could be made based on the fact that the Bible certainly does mention non-human intelligences. The difference between supernatural intelligences and aliens is something I've discussed in previous posts (such as Angels and Aliens), but nonetheless, the point could be made that the Bible clearly envisions intelligences other than that of mankind...However, the best answer to this question would be to point out that the Bible didn't mention the Americas either--yet they existed and furthermore were inhabited by intelligent beings--humans of course, but to people during the Age of Exploration it was a mystery how human beings had already arrived in this newly discovered land. Christians see no contradiction in the Americas not being mentioned in the Bible and existing anyway--they simply embraced the truth that while the Bible is true, it does not contain all truth that exists, that is, it does not contain all the information in the entire universe, nor was it ever intended to do so. For example, I, along with most of the readers of the Bible throughout history, am not specifically mentioned in it by name, yet I'm reasonably certain I exist ;)--so the Bible not mentioning aliens in the way we understand them in modern times is insignificant.
O2. The Bible says mankind is "created in the image of God." So if we are in God's image, anything else that does not look like us would not be in God's image. So no other form of intelligent life can exist.
A2. Er, this one is contradicted by the Bible itself whenever it describes angels differently from humans, such as the seraphim of Isaiah 6 (for a view of the seraphim that relates to this topic, please see my previous post The four faces around the throne of God--faces of aliens?)...they are intelligent, but don't look like us. And why couldn't aliens created by God be intelligent, but not look like us? Why would they necessarily have to be in the image of God? And who's to say that they wouldn't be in His image, even if they looked different from us? Perhaps the "image of God" is taken a bit too literally by some. God is able to see and hear...we have eyes and ears. God is able to move and we have legs, He creates and we have hands. God is aware of himself and plans for the future, and we human beings, in His image, do the same sorts of things...if that's what's meant by the "image of God," this is a trait we human beings could well share with extraterrestrials, if there are any.
O3. The Bible teaches the Earth is the center of the universe and if there are aliens, clearly their existence would show the Earth is not be the center. So nobody who believes the Bible should believe there are aliens.
A3. First off, not all Christians take the Bible literally at all, but among those who do (including me, except for clearly poetical or figurative parts), I don't believe we would agree the Bible teaches the Earth is the physical center of the universe--the Bible does not in fact talk about the universe in terms in which it makes sense to discuss a center. It simply says, "the heavens and the Earth"--the world we live on and the sky that surrounds us. True, Psalm 93:1 says "the Earth cannot be moved"--which doesn't say it's the center of anything--and it doesn't even say that the Earth "does not move," but rather that God has established the world as what it is and no one else can change that i.e. "move it." There are other passages, mostly in poetic sections in the Bible, but also in famously Joshua 10 ("the day the sun stood still"), which talk about the movement of the sun across the sky. First off, these passages are quite few in number. Second, they are from the point of view of the observer on the ground--and yes, I myself see the sun move across the sky. It is true that some theologians in the past used the Bible to justify the geocentric system of the universe devised by certain Greek philosophers...and who ignored certain passages of the Bible that did not line up with that system (including the mention of innumerable stars, which the Greek philosophers did not believe in, because they counted all the ones they could see and were certain there were no more).
My first answer was a little bit unfair in a sense, because even though according to what I understand, the Bible does not actually state the Earth is the physical center of anything, it nonetheless does clearly put our world at the center of a spiritual story. And why shouldn't it be--the Bible is the book for US after all, we human beings. But look at it another way--an implication of the Theory of Relativity is that all points of view of all observers are valid--time and space are variables affected by velocity and mass...which means there does not exist any absolute grid across the universe, nor any completely universal time clock, which means that each and every individual place is a much the center of the universe as any other. So why would Christianity be challenged in any way to find out first hand that perhaps aliens would see their own "center of the universe" as being every bit as important as we see our own?--our "heavens and Earth" without further specification would be just like their own view of their heavens and home world...And Christian theologians have long stated that God is Omnipresent--which means He is everywhere...in the center of every single place, throughout all space and time.
O4. The New Testament says Jesus is the Savior of the world. That would mean that He is not the Savior of any other worlds. Why would God save only human beings and no one else--it would not make sense for God to create aliens if Jesus just died for this world--so there must not be any aliens.
A4. The first two sentences don't follow logically as per what was said in A1. Just because Jesus is the Savior of our world and no others are mentioned, it does not stand to reason He is only the Savior of all the worlds (if there are more than one). Besides, who says aliens need saving? Perhaps any aliens which exist are themselves not sinful. That is, they could have a sense of conscience they perfectly follow at all times. This is a possibility that C.S. Lewis worked through in his space trilogy, especially the first two books. Or perhaps they are demonically evil, consistently violating their own sense of right and wrong at all times, and like demons, disinterested in repentance (mwaa haa haaa--I feel a story idea coming on!).
By the way, the New Testament Greek word for "world" in John 3:16 (as in "For God so loved the world...") is the word, "Kosmos," which, yes, you guessed it, can mean "universe" as well as "world." So John 3:16 could be read, "God so loved the universe, He gave His only Son..." What if the only Son manifested himself in the form of an alien on alien worlds?
O5. The New Testament makes much of Jesus being of the same sort of being we are--a descendant of Adam, which makes Him suitable to die in our place. Obviously he could not be the same sort of being that aliens are, so He could not be their Savior, so God must not have made any aliens (because that would be cruel).
A5. First off, that assumes aliens would be sinners, which they may not be, as addressed in the question above. What "sinners" means is having a sense of moral conscience, being aware of violating this conscience against your own will at times--that is, a sense of sin and a need for repentance and forgiveness. If humans encounter aliens and find that they like us are "sinners," I think it can be safely said that Christian missionaries will immediately want to preach the gospel to them. And if these aliens accept Jesus as their Savior, it would stand to reason people will say that Jesus being human was important in spiritual terms, not in the literal physical sense.
Secondly, who is to say that aliens would not have their own story of a Savior who died for the sins of their world? The Christian objection to what I just said would probably center around passages like Hebrews 10:12, which plainly state that Jesus died once for all sins for all time. So clearly He could not have died here and then later (or earlier) died on an alien world...or is that just talking about Jesus dying just once? If so, would an alien equivalent of Him count? And what if the Word made flesh (as Jesus is shown to be in John 1:1-18) were in fact actually the same being for all races of beings, human and alien alike, the same spiritual reality with differing bodies--could it be that all versions of the single Savior would all exist at the same time, live at the same time, and die at the same time, in effect, dying only once, even though simultaneously in many places? (after all, God can be everywhere at once, so why would not the Savior be able to die in more than one place at once? even though that is not what we would normally expect)
O6. Alien encounters described by UFO believers sound much like Medieval encounters with demons. Since we know from the Bible that demons are real, that means UFOs are fake and the so-called aliens involved are fake--these are actually demonic encounters!
A6. Uh, maybe. But even if UFO encounters were generally demonic, it would not necessarily follow that all of them are demonic, would it? And even if UFO encounters were all demonic, it wouldn't necessarily stand to reason that there are no aliens. It would simply mean the UFOs don't represent the real aliens that may actually exist on other worlds, beings we have yet to encounter. This opinion on UFOs actually has nothing to do with whether there are aliens or not...
By the way, I don't know if aliens exist or not--I don't think there is any way I can know without actually meeting one or some other form of direct evidence. It's interesting to me though that some atheist friends of mine are utterly convinced aliens must exist...even though they state they are atheists due to a lack of evidence of the existence of God...
O7. The New Testament has a story of the end of time (mostly in Revelation, but based on Daniel, Isaiah, Zachariah and other passages of Hebrew Scriptures) that is too soon for there to be any time to find aliens. And no aliens are mentioned there. So there are no aliens--or we human beings will never meet them, anyway.
A7. For a previous post on "aliens" in Revelation see my post Alien God of the Christian Rapture. As for the rest, well I believe the Scriptures deliberately put the Christan believer into a state of being that continually expects the return of Jesus at any time...and I don't think that it's been an accident that it's been so long. Yet, if it's been two thousand years, why couldn't it be twenty thousand years before the end? Granted, there are a number of things in these prophetic passages that sound very much like modern conditions to me (especially Israel literally reestablished as a nation, true since only 1948)--or sound like certain interpretations of the passages, I should say. Yet history shows that same sorts of things can happen over and over again in human events...human beings could spread out over time to many worlds, meet many aliens...and then undergo a long slow collapse back to just one world, our own...reproducing a number of conditions familiar to Biblical students of end times, twenty thousand years from now. And then the end could come. In short, we just don't know how much time there is. So in terms of time, human beings may well exist long enough to meet aliens someday...or perhaps we will only meet them in eternity, not mentioned in the Bible, but included among the things that the human eye has not seen nor the human ear heard, but which God has prepared for those who love Him (see I Corinthians 2:9).
So, aliens, which have become a common piece of modern American culture as much as zombies or vampires or superheroes, but which are thought to really exist by some very serious and intelligent people, should pose no real challenge for the Christian writer who wants to represent a worldview consistent with Christian doctrine...yet one that still includes extraterrestrials...