If these machines are capable of replacing us, then why would they care about us or do anything to keep our species going?
The only solution I can think of is Transhumanism: We need to stay ahead of the curve and remain the smartest beings around by augmenting. Those failing to do so will go the Malthusian way.
Where would a guaranteed income come from?
In the short run: Through taxes and money printing. BIG is an economic impossibility in the long run, however. The part of the population on BIG would separate from those who create the wealth. It would lead to an Eloy/Morlock situation except that the Morlocks have no use for the Eloy and therefore would get rid of them.