So in net you're not actually giving that $1k/month out to everyone, only the people who don't otherwise make anything. Then someone who makes $1k/month from other sources might pay $500/month in tax, thus reducing their effective, net UBI payment. (note, I made up these numbers completely off the top of my head)
Someone who makes much more than that would also pay much more taxes.
So in the end, UBI needs to be accompanied with some sort of tax reform, and isn't just about sending out checks to everyone (including Warren Buffet) on top of the existing system.