Enforcing the border is cheaper than all the other enforcement activities. That should come first.
I would not say that. The IRS can do a computerized match of names and SS numbers and catch several million illegal aliens in a few hours.
All they would have to do is tell their employers that they will be fined if they don't come up with valid numbers. The illegals would lose their jobs, and many of them would leave on their own.
That sounds pretty cheap to me. It is better to remove the incentive to come here and let economic forces control the flow. If no jobs were available, no one would try to sneak in.