We don't have to prove Aliens exist for the same reason we don't have to prove God exists. If we want to believe, we can. If you insist they do not exist--not simply by stating that, "I faithfully believe that they do not exist," then you must offer proof. Otherwise, you are guessing just as much as we are, and exercising personal faith.
I don't know whether aliens exist or don't exist. I crave tangible evidence. But I want to believe, and I'm okay enough with that.
Of course you can believe whatever you like, but for it to have any meaning outside of your head, you need some kind of proof.
I don't need to prove something doesn't exist, because I'm not really claiming anything. If that's the way the world worked, people would have to prove their innocence, or prove they don't owe someone money, or anything else anyone claimed. The burden is always on the person making claim. I'm not saying that they do not exist, I'm saying there's no evidence that they do, and I see no reason to assume they do until some is provided.
