If you are in the US - ask them to provide proof of a 5013(C) designation. If they do not have this document, the government DOES NOT recognize them as a non-profit entity.
Depending on the level of your interest, I would also ask to meet the board of directors - or at the very least, get their background information.
A quick search on Google usually provides plenty of insight on the "legitimate-ness" of an organization. Check facebook to see if they have a page, if they update their website frequently, have a physical address (look it up in Google maps).
If you do all the above, you will probably know more than necessary to make an informed decision. Hope this helps!