It would probably drive some up and some down.  People who have very little money have no bargaining power so they must pay what the landlord demands even for substandard accommodation.  Such accommodation would probably go down in price if a basic income were instituted at a reasonable level (high enough for a reasonable life rather than just enough to avoid starvation) because people would be able to afford something a little better.  This of course would drive the price of the bottom end of the good accommodation up.  But the equilibrium would probably be at a point that was more comfortable for the tenants and landlords would have to work harder to attract them.