http://www.politifact.com/truth-o-meter/statements/2010/aug/06/lindsey-graham/illegal-immigrants-anchor-babies-birthright/
According to the 14th Amendement, yes, they are US citizens by birthright.
If you'd wish to argue whether that should or should not be the case, though, that's a different argument entirely. There's a lot of fear of immigrants crossing the border to have children so that they can't be deported, but if you read that article that's not really a significant concern the way the laws are set up. An "anchor baby" has to be 21 years old to sponsor their parent(s) for citizenship, and if the parent(s) entered the country illegally, they have to return home for at least 10 years to reapply for citizenship, so it's not quite the drain on resources the right side typically makes it out to be.