I prefer Jerry Pournelle's solution... Since it costs an estimated $10,000 per illegal in govt handouts and direct costs to capture, detain, and deport each one, offer a $5000 bounty for each live illegal immigrant turned over to the border patrol at certain designated border crossings.
Or start the bounty at $1,000, but increase it by $500 each time a repeat offender is turned in. So the first time an illegal is caught and turned in, the bounty is $1,000. The second time that same illegal is caught, the bounty is $1,500. The US govt is obligated by law to track who enters and exits the country (legally or otherwise) so fingerprinting or otherwise positively identifying the illegal immigrants is perfectly legal. In fact, the govt is obligated by law to track people who cross the border, so it's pretty much a crime to NOT get positive id of all deported illegals.