Not sure what you refer to as "anchor babies". Immigration law only allows children to petition their parents for legal residency when they reach the age of 21. If the parents are illegal (and they actually get caught), I have never heard of them not being deported because their child was a citizen. There might be "bleeding hearts" out there, but neither USCIS nor ICE listens to them.

I agree that the children are not at fault and should have the benefit of birthright citizenship, notwithstanding the parent's legal status.