Isn't that a reason to lobby for "open" borders?
I'm down for more open borders, but believe that social benefits should be limited to minimal cases. Would also prefer to bring in more immigrants to work for understaffed industries, and would like to make it easy for the smartest people in the world to immigrate as well. In both cases they should have a clear path to citizenship.
In regards to illegals that aren't getting deported anyways, make them work for understaffed industries, become legalized, and start paying taxes.