A stringent green card system that gives permits to workers for a specified period of time.
A stringent penalty for lawbreaking employers is the answer. Mister Tyson, of Tyson Chicken for example, probably wouldn’t want to spend a year in jail for hiring illegal chicken pluckers.
I am not against that as an augmentation but you won't get that through Congress on its own. There are already penalties that aren't being enforced. My plan simply calls for using existing law.