Or better yet, put the senior executives of any company found hiring illegal aliens in jail!
I guarantee you, one high profile perp walk and the problem will be solved.
The fence has worked pretty well for Israel.
When I was hired as a technician for a small/medium size dealership and going thru all that process, it was so stringent that I thought I applied for a job at NSA nevertheless.
The problem is with rogue construction/roofing/grading companies and people like you and me hiring those cheap hands.
I pass by every day by a designated "laborer site" and it is filled with Mexicans. Somebody must keep them happy!