It depends on the size and the stability of the basic income. If a bank knows a customer will get $X/year for the next 5 years, giving an upfront loan to allow that person to create a business becomes a lower risk proposition. A basic income could easily expand the number of low to medium-risk borrowers and expand the amount of savings present to use as a base for lending.
I'm not an economist so I'm sure there are variables I'm not accounting for, but here's my opinion. I think the difference is that an endless supply of money (we know it's not truly endless, but a large influx of cash) allows commerce to occur and people can buy/sell more than commodities. The mere existence of a bank doesn't improve the lives of citizens if said citizens have no money to purchase food/goods and no assets to offer as collateral to take out a loan in the first place. There needs to be some sort of economy in place before a bank will benefit a population anyway. The bank could also benefit because residents now have assets to leverage when seeking cash above and beyond their basic income since I'm assuming starting a business may cost more than donations or a basic income would provide.
> an endless supply of money (we know it's not truly endless, but a large influx of cash) allows commerce to occur and people can buy/sell more than commodities
I'm not an economist either, but I think the concepts you want to look up are 'money supply' and 'velocity of money'.
I don't think this works. Presumably cash has a lot more utility than rice or shoes, for example. These people can always use more money (i.e. bank loads in your example), but once they have x shoes or y rice, the utility of new shoes or rice is very low.
> Presumably cash has a lot more utility than rice or shoes, for example.
It has a lot more fungibility, which should also mean that the marginal utility of cash drops slower (as a function of total utility from that source) than for rice or shoes.