Why would they? Would people in general be "earning" more money from basic income than current benefits? I mean you can't increase rent past the point people are able to pay.
You can increase them past the point that people on lower wages/basic income can pay, driving them out of desirable or semi desirable areas to cheaper areas. See London as an example.
Like I could move to Helsinki if I wanted to on my current salary, but I see no reason to do so. I live in smaller city (or maybe it's even considered a town) and I have plenty of extra money left over after my rent.