This could just be solved by having a total BI payout pegged at a fixed % of overall tax revenues. As more people earn less and pay less taxes, BI payouts also decrease as they are split among more people so the burden on the remaining taxpayers doesn't overwhelm them.
Then we could adjust that % every year (or not) based on actual conditions, how generous / prosperous / stingy we as a society are feeling, plus it gives a mechanism by which automation productivity gains can get rolled back into a BI program to (warning: utopian thinking ahead) eventually fund fewer workers overall, plus workers who work fewer hours, but we still have economic/automation gains rolled (via ongoing taxation) back into the BI support.
This basically shores up the "moral hazard" counterargument to BI proposals (which is the first thing that comes up for me after the "how do you pay for it" thing Max solved).