You really don't have to "close the borders" to control immigration. All you have to do is make it easy for employers to log in to a website and verify thet the person standing in front of them is a citizen. By a photo and a SS number. The employer can print out the report and file it. If INS ever checks the employer he will have the report on file. No ID cards, No green cards, just an on-line database of photos.
Then if any business breaks this law, throw the book at them.
Simple.
Of course I would require schools and welfare agencies to perform the same little task. If any bureaucrat is caught violating this rule, they should be fired on the spot.
Technology may hold part of the answer. But I believe sealing the borders shut and using the military to accomplish that goal, is a major part of the equation to solving the illegal immigration problem.