That's the opposite of what's being proposed: the minimum income comes from the government, not employers. Minimum income can replace minimum wage.

For example, say I'm a landlord. If everyone all of a sudden had an additional "base" income, why wouldn't I increase my rents to absorb at least a portion of that? Then, only people who had a job would still be able to afford to rent from me, while those on the basic income would be unable to afford it. I'm not out either way.

Sure, but now that you've raised the price, there's an opportunity to undercut you that didn't exist before. The result ultimately depends on the supply and demand of housing. Just think of it like a bunch of slightly-richer people moving into the neighborhood. Prices will rise, but not enough to take all the new residents' extra money.