That's why BI needs to be restricted to citizens. Not to legally here, not to illegally here, but only to citizens, with proof of citizenship. And only to minor children of citizens (and no others), to nix the anchor-baby problem.
I'm not much for socialist welfare and such, but I do think BI is better than the current impossible tangle. Get rid of the entire welfare state (with the possible exception of some medical aid to low income) and replace it with BI, and it would probably cost taxpayers less and do more good.
One of the problems with the current system is that it enforces poverty, because if you do save and invest what little you've got, or own your home, or for one disabled payment program, so much as own a car -- you're no longer eligible. I know people who stay poor on purpose because otherwise they'd lose their benefits. I myself have been dirt-poor/homeless yet ineligible because I'd socked away my paltry little inheritance as the only retirement fund I'd get, rather than spending it right off.