We should make the effort, on principle, just like we should have a "war on drugs" on principle. But in the end, both are dependent on the DEMAND side, not supply.
I think a guest-worker program would do a lot to end the demand side, especially if you kick in some favorable consideration toward citizenship at the end of x years working. That would a) increase the number of legal workers who have a stake in keeping out the illegals, and b) provide "fairness" to those who want some chance to become citizens.