We could start by not penalizing them for trying NOT to hire illegals.
(YOU CAN'T CHECK IF THEY ARE LEGAL UNTIL YOU'VE HIRED THEM! is quite possibly the most pernicious law we have.)
Could make a pre employment e-verify system, where citizens and visa holders are given a different number from ssn. This number has to be on all job applications, that way employers can check via a .gov website before even starting the interview/hiring process. Hell even drivers license # would work.