When the first soccer mom gets nailed for hiring day labor, you're going to have a PR disaster on your hands.
People are against illegal immigration in the same way that they're in favor of motherhood and against sin. It's a generically accepted ideal.
Doing what you describe would inconvenience the American public, and that will go over about as well as a brick balloon. Generic ideals tend to fall apart when unpleasant specifics come up.
Welcome to America, where everyone wants to go to heaven, and nobody wants to buy a ticket.
Punishment should fit the crime mom pays a $100 fine. No PR disaster here. Wal-mart pays a million dollar fine. No PR diaster there either. You asked for solutions I am giving you solutions that will work.