Realistically the only option we have that might work is shit-canning most welfare and incentives for non-productive immigrants to enter and make it pointless to pop out the kid unless you have a plan to make both them and yourselves productive members of society. Illegals showing up and popping out a kid and getting free WIC, claiming (stealing) the newborn citizen's welfare benefits, public schooling, chain migration via anchor baby etc are all going to have to be fixed through congress during some fluke period when the filabuster can be overcome.
If the clouds part and Jesus himself descended from the heavens, you'd ask him to ... discourage anchor babies? Surely there are better and more pressing uses of his power.
Part of the reason why immigrants were so successful and beneficial in the 1870-1920 era boom was that labor was so badly needed in the burgeoning age of industry. But really, the other half is there was no other option -- you did something productive or you were completely fucked.
There is a somewhat intermediate of this, proposed by Milton Friedman, called the negative income tax. I can't say I'm sold on the idea but it does solve some of the problems of the local maximums encountered that keep people trapped in the welfare system and from trying to get more lucrative income.