From Slate, an argument that addressing the problem will require a constitutional amendment:
"Anchor babies" are back in the news: Be prepared for another round of railing against the granting of automatic citizenship to the children of illegal immigrants. There was a burst of this last summer, when Sen. Lindsey Graham rumbled about pregnant Mexican women crossing the border to give birth and win American citizenship for their babies—which he inelegantly called "drop and leave"—and how it was necessary to change the Constitution to stop them. Now Rep. Steve King of Iowa promises to end automatic birthright citizenship through legislation, and conservative legislators from five states are talking about excluding kids from a new thing called state citizenship, and also creating distinct (second-class) birth certificates for these kids.
Legally speaking, this is all pretty confusing.