The key to answering this question is to understand what it the different types of radiation do to matter [see here for more explanation]. What you want in vision is help in getting around the world. Brick walls, trees and other people stop you and hurt you should you bump into them, and so it would be helpful to view them as solid. Seeing in X-rays wouldn't be of much help, therefore. Conversely, air isn't much of an impediment at the sort of speed we tend to travel, and this should ideally be represented, visually, as not being there (i.e. appearing transparent). Although there are obviously going to be exceptions, the best bet would be if our eyes were sensitive to the part of the electromagnetic spectrum that penetrated through the things we could penetrate through (like air), but couldn't pass through the sorts of things we couldn't pass through (like rocks).
What I've just explained is complemented so beautifully by Carl Sagan that I can't resist quoting him at length:
We're prejudiced toward visible light. We're visible-light chauvinists. But if our bodies could transmit and receive radio waves, early humans might have been able to communicate with each other over great distances; if X rays, our ancestors might have peered usefully into the hidden interiors of plants, people, other animals and minerals. So why didn't we evolve eyes sensitive to these other frequencies of light?
Any material you chose likes to absorb light of certain frequencies but not of others. A different substance has a different preference. There is a natural resonance between light and chemistry. Some frequencies, such as gamma rays, are indiscriminately gobbled up by virtually all materials. If you had a gamma-ray flashlight, the light would be readily absorbed by the air along its path. Gamma rays from space, traversing a much longer path through the Earth's atmosphere, would be entirely absorbed before they reached the ground. Down here on Earth, it's very dark in gamma rays - except around such things as nuclear weapons. If you want to see gamma rays from the center of the galaxy, you much move your instruments into space. Something similar is true for X-rays, ultraviolet light and most infrared frequencies.
On the other hand, most materials are poor absorbers of visible light. Air, for example, is generally transparent to visible light. So one reason we see at visible frequencies is that this is the kind of light that gets through our atmosphere down to where we are. Gamma-ray eyes would be of limited use in an atmosphere that makes things pitch black in gamma rays. Natural selection knows better.
The other reason we see in visible light is because that's where the Sun puts out most of its energy. A very hot star emits much of its light in the ultraviolet. A very cool star emits mostly in the infrared. But the Sun, in some respects an average star, puts out most of its energy in the visible. Indeed, to remarkably high precision, the human eye is most sensitive at the exact frequency in the yellow part of the spectrum at which the Sun is brightest.
Might the beings of some other planet see mainly at very different frequencies? This seems to me not at all likely. Virtually all cosmically abundant gases tend to be transparent in the visible and opaque at nearby frequencies. All but the coolest starts put out much, if not most, of their energy at visible frequencies. It seems to be only coincidence that the transparency of matter and the luminosity of stars both prefer the same narrow range of frequencies. That coincidence applies not just to our Solar System, but throughout the Universe. If follows from fundamental laws of radiation, quantum mechanics and nuclear physics. There might be occasional exceptions, but I think the beings of other worlds, if any, will probably see at very much the same frequencies as we do.
From "Billions and Billions", pages 43-44