First, ASK

In the sign up, make sure your affiliates must affirm that they are 18 years of age. Ask for the date of birth.

There is not a way to check an ID online, but for US residents you should require a Social Security number. Since you are sending this person money, that is not an over zealous request. An affiliate that is going to send you good traffic will have a mailing address also. If an affiliate does not want to give you a mailing address, do you think that person is going to be a good affiliate sending a lot of traffic, or just a wanker fly by night??

At least make a conscious effort. You cannot protect yourself 100% but at least avoid the easy mistakes.