Both legal and illegal immigrants have very limited public resources available to them. This has been true since the Clinton era welfare reform.
Furthermore, it's not entirely clear that either legal or illegal immigrants are greater users of the public resources that are available to them, especially in proportion to native users.
You have two arguments here. One is welfare, which isn't as strong as you seem to think. The other is protection from social upheaval. Yet, it's always curious that it's the new immigrants who are creating a problem. Interestingly, the evidence that the most recent wave of immigrants, legal and illegal, from Mexico and other Latin American countries are causing much upheaval at all. Crime rates in the last twenty years have fallen. Some of the safest cities in the US are along the Southern border with high immigrant populations. The trouble doesn't seem to originate from the immigrants, but from native distaste for immigrants.
Now you're arguing for protectionism. Current immigration law tries to do this. But it fails miserably, possibly because the government does a pretty poor job of determining what the labor market needs. There is a better way. And in it's own odd way, it already works. Illegal immigration already regulates itself. It exists because there is a market for the labor. As the market shrinks (or expands), so does the labor pool. This takes place without any government supervision, yet it somehow seems to regulate itself. Why? Because immigrants, regardless of legal status, are not willing to relocate here unless they can improve their lot. The idea that the US would suddenly have millions of newcomers sitting around with nothing to do is a popular perception, but has no rational basis.
The easiest way to deal with this is to liberalize immigration so that potential residents can come, and leave, without any significant trouble. This makes the border more secure, because there is no incentive to cross anywhere but an official crossing point. Border patrol no longer wastes resources chasing aspiring gardeners and nannies. Instead, people crossing the border illegally in remote places can more easily be assumed to be up to no good.
The idea that so-called anchor babies is a problem is one of nativist perception. I've always been curious about the value of citizenship. Natives seem to think it's something so valuable, it should be rationed. Yet, I have a feeling if somehow, everyone secretly had their citizenship revoked tomorrow, half the people would never know any different. Citizenship offers few real privileges for most folks, and a few annoying obligations. It seems mostly a source of pride, a sense that new immigrants must show sufficient deference to America before we allow access to the club. Even if deeply rooted natives can be as dismissive, even hateful, of America and it's traditions as they want. Hey, it's a free country, after all.
Right?