It depends on what kind of aliens you’re talking about. If we’re talking about the “UFOs & crop circles” kind of aliens, there’s a very different set of arguments to be made than for the “intelligent life anywhere else in the universe” kind.
The arguments for life somewhere out there in the universe are mostly statistical. We are one planet around a very average star, and we’ve developed to become a rather intelligent species (or so we’d like to think)! It’s difficult to think of reasons why all the probabilities that aligned for our planet couldn’t have possibly aligned for any other planets. There are simply so many other planets, that even if the chances of everything lining up again in another solar system are very very low, by the time you account for the number of chances you have, it’s statistically mandatory for there to be other life out there somewhere. Even if the other life were so rare that you would only expect one civilization per galaxy, there are billions of galaxies out there - that’s billions of other civilizations.
That’s not to say that they need to be nearby. In fact, they’re probably not nearby. If we stick with our pessimistic view of one civilization per galaxy, then we ought to look to our nearest galaxy - Andromeda. Andromeda is 2,538,000 light years away. Two million years is an awful long time to wait for a sign of intelligence from Andromeda. Or if they’re watching us, they’ve got an awful long time to wait – homo sapiens emerged around 200,000 years ago, and our first radio telescope was built in 1931. It will be two million years before light from our galaxy produced at that time will reach Andromeda. So there’s a bit of a communication problem here. There’s even more of a transportation problem; while communication can travel at the speed of light, transportation can’t. There’s no way to make a massive vehicle travel at the speed of light - we have trouble getting things to travel even close to the speed of light, which means that if you were actually trying to go there, it would take many times longer than two million years.
We’ve been listening for signs of ordered radio signals from other civilizations off and on for about 55 years. This means we’ve effectively looked at a little bubble that’s 55 light years wide around ourselves. That’s a very tiny sphere of space around the earth, but considering the number of stars that fall in that sphere is not huge, not finding anything there is probably to be expected. SETI@home has been running since 1999, analyzing data from the Aricebo radio telescope to see if there’s anything of extraterrestrial origin in the data. The Allen Telescope Array is also meant to search for signs of radio signals from intelligent life within a much larger sphere of 1000 light years. It’s been running since 2007, and there’s nothing yet, from either SETI@home or the Allen Telescope Array.
If, on the other hand, we’re talking about UFO type aliens that people feel they’ve seen in the sky, all the above problems still hold. You’d still need to have spotted a civilization and managed to get yourself there without breaking any laws of physics. And then, once having reached a planet with an alien race living on it, you must be content to fly erratically in the atmosphere without trying to make any kind of further contact.
As far as I am aware, there is not a single UFO sighting that cannot be explained by either a meteorological effect, a satellite, high altitude balloons, other high altitude aircraft, or plain old hoaxery. This is not to say that people have not spotted things in the sky that are unusual - they very well may have - but an object in the sky that is not immediately identifiable does not give you proof of an alien encounter. Even if there are a few genuinely inexplicable things happening in our sky, it seems to me to be rather a large leap from “Something bizarre happened in the sky” to “extraterrestrial life”. Crop circles are entirely man made objects.
Most of the above points are relatively objective. If you ask for my opinion, I think it is extraordinarily improbable that we are the only form of intelligent life in our vast universe. There are so many chances for it to arise, it must have happened elsewhere. But my understanding of the that same vastness means that I also find it extraordinarily improbable that that intelligent life is anywhere near enough for us to ever contact them. The statistical arguments in favor of other life are as overwhelming as the statistical arguments that they are enormously distant from us. But do I personally believe that any of these advanced civilizations have found us and buzzed around in our atmosphere? No. The evidence for this kind of thing is shaky at best, and I find it hard to believe that a civilization apparently advanced enough to reach us would not have the curiosity to come attempt to interact with us more directly. I believe that most UFO sightings are the result of a combination of things happening in the sky that are hard to describe, and a lack of awareness of what you can see in the night sky.
Something here unclear, or have your own question? Feel free to ask!