restrictive government quotas on visas (i.e. legal immigrants), which are woefully out of synch with the free market for labor? In other words, it's essentially a problem created by big government interfering in the free market. The issues of illegal and legal immigration are inexorably intertwined.
If you were to increase the amount of legal immigration permitted to a level that's more in line with the free market, then illegal immigration would mostly vanish.