Well there is an easy cure for this Tsunami. There are few to no employers in the U.S. who would not prefer to hire a U.S. citizen over an illegal immigrant were the wage they had to pay the same. They hire illegals because they can pay them less and, usually, get away with it. The government does not vigorously enforce labor laws against employers, and when the employer gets caught, the fines are not sufficient to make the hiring of illegals uneconomical.
Why not make the hiring of illegals unattractive by raising the minimum wage to $10-$15/hr coupled with very vigorous enforcement of the labor laws. The cost of better enforcement can come out of reduction in border control expense. Then you'd see competition in the labor market. The illegals will be out of a job and go home. The low wage U.S. citizen worker currently competes badly against the low wage illegal worker. At present, those who must meet living expenses via minimum wage can't do it on their own, and some choose to milk the welfare system rather than work for a ridiculous wage. (Illegals are willing to accept a lower living standard and may be paid in cash with few deductions.) Those adult U.S. citizen workers that do choose to work for a low wage end up being subsidized by someone else, often the taxpayer.
The old saw that if we raise the minimum wage everything will adjust upwards by an equivalent amount, and we will be right back where we started, is utter nonsense. That's never happened in the past. Why would this time be different? Some have claimed that a rise in the minimum would result in a wage price spiral. They don't understand the factors that must be present in the economy for that to happen. Currently we have an exceptionally weak labor market! The indirect effect on prices from a rise in the minimum should be small because only a fraction of U.S. citizens are working below the new minimum target, the cost of low wage labor is only a small (often tiny) fraction of product cost, and as long as competition is present, a part of increased labor cost will be eaten by management. Mind you, the minimum must not be raised above the true cost of minimum wage labor, or a distortion in the other direction will be created and some of the predictions of the uninformed would then come true.
What's odd is that we have republicans resisting a minimum wage hike, ignoring weak enforcement of labor laws (you never hear either the democrats or republicans clamoring for enforcement against employers!) and spending all their efforts in a failing attempt to shut off the border at very high expense to the taxpayer. This is so inconsistent it's almost schizophrenic. The best solution to the problem of illegals flooding the low wage market is right in front of their eyes. If you want to shut down the flood of illegals, then remove their incentive! You'll save a whale of a lot on border control and create a much less distorted labor market. (The republicans are the party of business. Sadly the average businessman tends to be anti-intellectual and not capable of rational thought when it comes to the economy and government policy. Some of them even believe in supplyside, trickledown economics, despite this economic fad having long been shown not to work as envisioned. This propensity of business toward weak thinking is reflected in their political party. But please, however, do not interpret this remark as a ringing endorsement of the democrats.)
By the way, what I advocate would make a sensible "open border" policy far more feasible than does our present, illogical approach to problems in the U.S. labor market.