The fact that illegal immigrants are willing to work for less than minimum wage is just proof that the minimum wage is too high. I know most people don't want to hear this but it's a fact of globalization.
Eventually, the world's standard of living must equalize. This means that individuals in "rich" countries like USA will see theirs fall while most everyone else who lives on $2 a day will see theirs rise. We've known this was coming for a long time. It would take 4 Earths for everyone in the world to have the American standard of living.
The fix would be to lower minimum wage law or abolish it completely. Otherwise, jobs will continue to go to immigrants.