What kind of welfare? Do you have stats or stories? I'm skeptical someone wealthy would slum it for the peanuts they could get from welfare fraud.
Or if they do, surely there are ways wealthy "real Americans" could hide their money just as easily and defraud welfare, so this isn't really a problem specific to birth tourism.
They can do whatever you presume the rich birth tourist does, like deposit in foreign banks, not own US land, obfuscate land ownership through trusts or straw owners.