I don't like the fact that the U.S. is willing to spend $10,000+ on citizens' health care if they are poor, but unwilling to spend $20 on an African or Mexican's health care if they are poor. But that is how it is, so people are going to climb fences and break laws to get their kids the free health care, access to top universities, large financial aid packages, etc. (I'm not saying that's the only incentive--the $10,000 a year low-skilled jobs are a big incentive too--but it exists.)
It's fair to ask if we should eliminate birth-right citizenship since it seems to help a lot of poor Mexicans. I think the answer is, ideally, yes, because Mexicans should not be allowed to cut in line to get citizenship. Poor Africans, Haitians and other Latin Americans also want to come here and would benefit from free health care and $10,000 a year jobs. They'd also, I suspect, be more willing to learn English and integrate into society better. That's a win-win. But in reality clamping down on illegal immigration probably won't lead to expanding legal immigration, so I have mixed feelings, mostly against.
George Will discusses some not-so-important-but-lawyers-care-about-it history of the issue.