But you can rescind the anchor baby laws, beef up border security (including building a barrier), and heavily fine employers who hire illegals. Taken together those measures would go a long way to help solve the problem.
Some of them, but the citizenship status of the baby is a matter of the Constitution. That's not to say that you can't send the mother/parents home, and let them decide to take the baby or not, it's just that the baby will retain his/her US citizenship. I know a guy, and a much younger gal, that are in that situation. Both born to Mexican parents, probably illegal but I don't know, in the US. Went back to Mexico and only latter returned to the US. She has an associates degree, and is working on her bachelors, between working 1/2 time and caring for a young son and younger twin daughters. I've lost touch with him, and in any case he was only an acquaintance, she was a coworker.