Say we both operate hot dog stands on opposite corners of the street, each charging $1 per hot dog. Basic Income goes into effect and suddenly 2X as many people can afford hot dogs. You raise your price to $2 in response. If I have any sense, I'll keep my price at $1 or maybe lower it slightly. More people will buy from me, significantly increasing the amount of money I make despite the fact that I haven't raised my prices.
This concept applies to most sectors of the market. People will still be price conscious, and businesses will benefit as much from more customers as they will from higher prices. Even with housing. If more people can afford apartments, more people will decide to convert existing real estate or rent out rooms which will hopefully increase the price competition. This all assumes that there isn't open collusion between market participants of course, which there very well could be. But the general idea is that prices won't rise uniformly in every sector of the economy at once.