As far as I can find through research, the “anchor baby” theory is based upon interpretation of the 14th Amendment and the 1965 Immigration Act.
Both legal scholars and the courts are sharply divided on the issue of the 14th Amendment (that is, allowing for/against the legal status of anchor babies). The author of the amendment, Senator Jacob Howard, made it clear his intent was not creating citizens ship for the children of illegal aliens. “U.S. Senator Lyman Trumbull, chairman of the Senate Judiciary Committee, supported this view, arguing that “subject to the jurisdiction thereof” meant “not owing allegiance to anybody else and being subject to the complete jurisdiction of the United States.”
Considering this current interpretation of the 14th Amendment is both recent and likely very political in nature, I would not bet the farm on it being permanent.
The 1965 Immigration Act was just that, an act of Congress. It is subject to change whenever a majority of Congress desires it.
Where does all this leave the anchor babies? According to statute, at the age of 21 the child can invoke their right to citizenship. The can further, after fulfilling all necessary legal hoops, can sponsor other family members. The catch is at 21 they must claim and prove (a simple birth certificate is proof) their rights to citizenship. So, for the first 21 years neither parent nor child has legal status. Families have been and often are removed from US soil.
Being an anchor baby is both a long-shot at best (21 years) and certainly no “quick fix”. Children already born under this legal situation would likely be “grandfathered-in” under future changes, but there is not even the guarantee of that.