Could go either way, maybe more crime if they knew there were no gun owners. Also even if you did ban guns they would still be smuggled into the US through South America and Canada leaving only the criminals with guns still.
Funnily enough the only articles I can find on gun smuggling and Mexico are from guns being smuggeled from the US into Mexico and not vice versa.
From the US immigration and customs website:
'HSI and its law enforcement partners target the illegal movement of U.S. origin firearms, ammunition, and explosive weapons with the ultimate goal of preventing the procurement of these items by drug cartels, terrorists, human rights violators, foreign adversaries, and other transnational criminal organizations and individuals that utilize these weapons to facilitate criminal activity and commit acts of violence.'
Perhaps TV and films have misrepresented this but it seems the issue is guns leaving the US, not entering.
So illegal firearms most likely crop up from other means. As others have suggested, probably from stealing from legal sources inside the US. As I've said before, if you make guns illegal, owning and carrying them illegal then you cut of the supply of illegal guns. Eventually confiscations ect will clear up the streets for the most parts. Naturally there will be exceptions. But if there are only 230 justifiable homicides vs 8000 then who's going to notice? As no one is actually really using guns to defend themselves in the first place relative to those using them to kill criminally.