All you need to do is to verify that he is of legal age and, if possible, that he is not prohibited from buying or owning a weapon- felon, etc. Since you can't verify his criminal record that is pretty much a moot point.
I would copy his driver's license info if he will let you- some people are touchy about private info, and keep it for future reference- just in case.
Suppose that weapon gets sold again or stolen, used in a crime, and you are the last known purchaser of the weapon? If you have a dated bill of sale, or at least the buyers info, you can be cleared. If you just tell them "I sold it a year ago to some dude" they might not believe you.
If someone was buying a weapon from me and refused to let me have his name, address and driver's license number I would decline to sell to him.
Jim