You have two mechanisms by which you can fund such a proposal:
* The government prints money and hands it out. Basically, just direct inflation; the BI becomes an arms race to keep up with inflation caused by BI.
* The people who successfully make money have a larger portion of it taken away by the government and given to people who don't make money. They recoup their losses by raising prices and re-extracting it from the people it has been given to, except now the government gets a cut of it twice. BI has to be increased to keep pace with rising prices caused by funding BI.
(There is a third proposal: Borrow it without any plan to actually repay it, which is not self-sustaining for obvious reasons)
In either case, I can't understand how a BI would lead to a more stable economy. The endgame of the first option is Zimbabwe. The endgame of the second option is Marxist communism; the point at which the system reaches stability is the point at which there is nobody left to extract extra money from to distribute.
I feel like I'm missing something critical that makes it a viable strategy. I don't think anyone at all is in favor of uncontrolled hyperinflation, so I'm left to conclude that the favored option is the second choice. We know from history that communism just doesn't work once you reach a large enough population that people are not personally accountable to each other; why would it be different here?
(I want to make it clear that if I could snap my fingers and make the world such that nobody had to live in poverty and had food, clothing, shelter, and an internet connection, I'd do it. I don't have anything against the moral goals of BI. I just don't see how it's sustainable, since the money for it has to some from somewhere.)