Present-day welfare systems have tremendous amounts of overhead to run. In a basic income system, most of that overhead goes away - you don't need an Office of Unemployment Benefits, or a Food for Underprivileged Children program - because everybody gets a check.
Do the savings completely offset the cost? I have no idea. But, it seems reasonable that they could offset a substantial portion. Of course, getting a government to willingly reduce it's size is a whole other problem.