Ban person-to-person gun sales.
The thing to bear in mind here is this: The US has so many legal firearms, so many of which turn into illegal firearms, that people are smuggling them to Mexico!!!
Chew on that for a sec.
If people are smuggling something from a high-earning country, into a low-earning country, there's something not right.
The FBI is confident that theft, gun shows and person-to-person sales account for the bulk of the illegal weapons.
Considering that gun show salesmen are allowed to sell firearms without background checks for the same reasons private persons are, that's a two-fer.
As for theft: Put damn license plates on the weapons. Demand secure storage. Theft is hard to avoid, but it is possible to make people store the weapons so that they're hard to steal.

Illegal guns are a big problem, and there are insane amounts of them.