Ahh the notion of invisibility, strictly speaking for something to be visible (to us) it has to interact with visible light either by reflecting it, or absorbing it. If it does not poses the ability to interact with light is it effectively invisible. However, things that are truly invisible may still interact with other things which can then be measured.
However this is not what you mean is it? Would it be correct to assume that your question is more in the line with :
"How can you prove something that is seemingly non-existing"
Physics has tried to tackle questions like this multiple times, and the consensus seems to be that they are really really hard to prove! heh
The first thing you want to be doing is to figure out how, whatever it is your trying to prove, interacts with the world. Once you got that down, you can set up an experiment that in theory would show if your "THING" interacts in the predicted fashion. Hence indirectly proving its existence because of its interaction. Worth noting is that you need to keep in mind that you are only proving the interaction, which may later be proved to be caused by something else.
Areas where this line of thinking is applied at the moment, finding the higgs particle, finding the graviton (Heh, we still dont know what causes gravity )
Did this help at all?
Kind regards Mads