“Problem” scare quotes as intentional ridicule.
Easy to solve. You make border travel totally legal, and you have a special wage, maybe half, even a quarter, minimum wage, call it the, “limited work visa wage.”
Now, if they try to get jobs using that cheap wage as an incentive, their own people who have full citizen wages will shut them out. Boom. No more border cops. No more 4th Amendment threats. It cuts all the corruption out of the system. The people police their own to keep wages up. “The Man,” in both Mexico and the US, loses.
I am Diabolical, yes. It would work.