Brenda WalkerÂ refers to "BirthScam Citizenship" below, and we do need something else to call it besides "Birthright." Perhaps " Stolen Citizenship" would get the idea across better.
Steve Sailer wrote 
"Birthright citizenship" is a phrase loaded with good connotations. People tend to react well to the words "birth", "right", and "citizenship". (Consider their antonyms-"death", "wrong". and "treason".) The Boy Scouts award three merit badges with the word "citizenship " in the name. The Scouts emphasize the duties and obligations beholden in citizenship. That's what "citizenship" sounds like to the average patriotic American: doing right by your fellow Americans. They need to be reminded that, to the Zhous and Chaos of the world, it's just a sweet scam.
When you need a technical term for the current state of law,  call it "jus soli ". That's Latin for "law of ground". Nobody has a warm and fuzzy feeling about a couple of words they aren't sure how to pronounce. (It's pronounced yĂĽs-so--le-.)
Similarly, let's use "anchor offspring" rather than "anchor babies". Babies are cute."
And as, as Steve goes on to point out, babies go on to grow up. Some one who was an adorable little baby in 1995 may be a full-fledged gang member today.
There's actually a teenage girl gang in Oregon called Latin Babies. [La Vida Loca - Hispanic street gangs take over Portland's east side.  By Nick Budnick,Willamette Week, June 30, 2004]They're not as dangerous as a teenage boy gang, doing mostly vandalism, the odd break-and-enter, and fistfights. (With other girls. )
But unlike Latin Babies born south of the border,  they can't be deported, even if one of them gets hold of a gun or knife and seriously hurts someone.
That's because their parents, who wanted a better life for their children, succeeded in stealing citizenship for them. But it's not their birthright. It's yours.
Send any suggestions for new names for this phenomenon to firstname.lastname@example.org.