My parents paid taxes for me to receive my education, and in turn so did theirs, and so on. A person that crosses the border to give birth in an American hospital, brings their children to American emergency rooms, has their children attend American schools, etc uses far more resources than they pay back. Our system is built such that when we do bring in new citizens, they're largely educated, skilled, and/or wealthy, so that they're a net gain for our nation. Not nearly as likely when we're talking about unabated flow of cheap labor and its progeny, particularly not for those that first give birth, and then find work.
In the end, the children still benefit massively by being born here. Whether we're talking someone born in America then raised in Mexico, born here and then leaving with their parents halfway through childhood, they all still gain the right to live here as adults. Their lives are made worse only in the sense that the life of a person exploiting a mistakenly-cheap Amazon/Newegg deal is made worse when the correct value is entered.