I don't think that's how rent is set right now. I believe people charge rent based on what people can bear to pay - so as long as another person with BI comes along willing to pay the higher rent, the landlord will sell to him instead!
It's not the house that's valuable to a renter, though. Most of the price variation for a given floor space comes from location-driven land cost. If someone on basic income plans to live solely on their BI for a bit, maybe to work on a personal project, maybe just to play videogames, they won't be location-bound and can seek cheaper housing in the suburbs or a rural area. There is a lot of underutilized land in America that could become very attractive to developers if the pool of available consumers suddenly included a subgroup that didn't care about commute time. That's not just greenfield developments, either; it would also apply to major cities, like in the Rust Belt, where housing and land are incredibly cheap because there are no jobs available.