You can test for the implications of the subject's existence.
For example, if you're going to prove there is no atmosphere on a planet: You know an atmosphere implies that it would be possible to gain lift in that environment, so you could run a wing over the surface of the planet and measure the lift.
If there is no lift, the implications of the subject's existence are not fulfilled, hence the subject does not most likely exist.
[sub]yeah, I know it's a pretty bad example, but I can't think of anything better right now.[/sub]
If you're talking about proving something doesn't exist at all though (In the case of the atmosphere example, that would be proving no planet in the entire universe has an atmosphere), you're in for a tougher job.
You could still go about like in the first example, searching for effects that are necessarily produced by the subject, were it to exist, but the universe is infinite, so experimentation can only get you that far.