I was asked this as an undergraduate once by the editor of a college termly - Oxford Daze - which then turned into me being asked whether one of the other students was an alien. This was written up as:
After a few well-aimed kicks to the groin, Mr Pointer admitted his previous position was untenable and [ ] was an alien.
Often I am asked more seriously - and the answer I give is that I think it is unlikely, but I could be proved wrong. I also note that, if aliens exist, we do not know in advance when First Contact will be. We will not even know the second before it happens. It will be totally out of the blue.
And no, I don't believe aliens are kidnapping people for medical examinations. Any alien race advanced enough to travel to Earth would be advanced enough to hack the NHS computer system and download all the information they want.
I also need to add that my reason why I think aliens are rare is not a theological one. After that incident with the Martian meteorite that was thought to contain traces of life (it didn't), there were voices on one side saying the Bible teaches that aliens don't exist (it doesn't) and on the other side that discovering aliens would bring religious belief systems crashing down (it won't).
Sometimes my stance is objected to with the "Johnny Alien" logic - an extension of the Johnny Foreigner approach. For example, maybe aliens prefer not to travel in space, maybe aliens have such different technology we can't detect it, maybe aliens choose to leave us alone as we are not
edible advanced enough.
The problem with this is that it would have to apply to all aliens - if just one alien race chose not to leave us alone, then we would have First Contact. If just one alien race nearby had technology that emitted at radio frequencies, we would detect it. You would need a galactic equivalent of the United Nations, with treaties and suchlike, with not a single rogue state, not a single explorer who wants to visit that pale blue dot, and not one private exo-entrepreneur thinking that our resources would be ripe for exploitation.
I have just seen the question Do all living organisms in Earth share one ultimate common ancestor? Or did life "begin" more than once in separate places? . And this is relevant to my stance - an argument I have seen and believe is convincing.
Suppose that life takes hold where it can. So if an alien planet could support even basic life at any point in its existence, it will. The universe is bursting into life, and ET is common. This would then apply to Earth. Wherever life on Earth could appear, it would.
And it would have done so many times and in many places.
This would mean that rather than one common ancestor, life on Earth would have many, with different lines which can be traced back to different points where life appeared. Yet we don't find it. Not one microbe, not one bacteria, not one animal with which we have no common ancestor.
This suggests to me that life starting and taking hold is a rare event.