Which would become a great excuse for dumping victims of all sorts. What's needed, as in any plan, are reliable men. You need set of rangers, who will not shoot first, but who will defend the border as necessary. Maybe a federal agency. Maybe a pseudo-private agency. Fold the INS border police right into that and take the best of them.
Doesn't the US have bases in Mexico? If so, that would be the ideal processing station for repatriating such foreign citizens. I'd hesitate to suggest a bounty/live head since it might encourage those to snatch people and claim they were border-crossers who were not. But if run by responsible and duly sworn rangers, well trained, with full military support all along the border from CA to TX, and with processing on US soil in Mexico itself, I think it might help make the point and stem that tide.
I would rather have rangers who would shoot first. After a month or so they would notice a scarcity of targets. And ammo is cheaper than repatriation process.