1. It's not going to be an amount barely above the poverty line. If it's implemented "correctly" it needs to be enough to do away with all the other welfare programs. Think way, way higher than $12K per person per year.
2. With less revenue coming in, guess who's going to be hit with higher taxes? All those suckers working to make a marginally higher income.
3. If the BI is sufficiently high (think in the $45K range), there will be a lot of people leaving their jobs to live a life of leisure and avoid having their income taxed at a >50% rate. I know I would.
This utopian idea is like every other: sounds great on the surface, may even work reasonably well at a small level, but doesn't scale. Human nature is what it is.