It is, ironically, a situation where the best solution might very well be a lighter touch.

US immigration policy used to be simply... Less. You could pretty much come and go as you please along the southern border. So people did, and therefore there were simply fewer issues because folks didn't feel the need to "sneak in," not when getting in and out of the country was as easy as a day-pass. People would live in Mexico and work a job in a border city.

The tightening of the rules began in (if I recall correctly) the '80s, and with it the actual crisis; if you could come and go as you please, it was no issue to live in Mexico, but people, when forced to choose, stayed where the economic productivity was.