You need to look at this in the "end game" scenario, which is, we need BI because there will be so much automation that hardly anyone needs to work anymore. You're not going to sell this to the poor by telling them that they're now unemployable and their BI is going to be less than their previous income + food stamps + housing subsidy + etc. So right off the bat, yeah, the BI floor needs to be the cash value of a person with full welfare benefits. And that's a lot more than $12K.
You're also discounting the inevitability that politicians will promise bumps to the BI for votes, people protesting that they can't afford to live in Manhattan on their BI, etc. It will quickly reach a point that even skilled workers will say "F this" and join the ranks of the unemployed.
Assuming your household consists of 5 kids, your spouse and you. 7 people x $12K/person = $84K. That would be 2x the average family income in the US today!