NO one lets them in, they usually sneak in, or overstay their legal work visas. It just doesn't happen. The ones caught are deported.
The problem can be solved by reducing companies incentives for hiring them. Increase those in positions of ie; police, doctors, teachers..etc. expanded authority to report suspected illegal aliens to ICE.
I don't see this happening until we solve our first problem...getting rid of incentives. Those same hospitals treating illegal aliens with medical care, often free....are the same hospitals hiring illegal aliens as orderlies and nurses. The same goes for education...as we provide free of charge, an education to illegal aliens, school districts hire illegals as janitors, landscapers, dishwashers, food handlers...etc.
The entire top of the system has to change, or nothing will change. Minutemen can waste their time all day at the border, but nothing will stop "cheap labor.