Look at the most basic need: food. Demand is relatively constant - you need enough to eat, and shouldn't eat too much (obviously in practice there is a certain amount of elasticity and differences in demand for different types of food). Supply of food already exceeds demand (look at food waste, supermarkets tossing out ugly produce or burning unsold items to maintain their profit margin). So - demand for food barely goes up, supply continues to exceed demand.
I would say shelter is as much of a basic need, and places like the bay area show that prices are rapidly increasing due to demand. Would a basic income exacerbate the problem in places that already have an issue with high housing costs?
As far as food goes, wouldn't the prices of luxury foods, like porterhouse steaks and the cost of going out to eat likely increase rapidly? I mean, no one needs to eat expensive food, but isn't the purpose of the basic income to prevent people from having to eat ramen 3 times/day?
The bay area is an interesting example. Prices are skyrocketing from demand, but part of the reason why supply has been so slow is that demand has been so sudden. San Francisco has always had a high quality of life, but it did just fine until the tech booms. The majority of people now moving to SF are coming for economic reasons. Would UBI reduce these motivations, easing pressure on the local rental market? I don't know.
SF definitely demonstrates that the price of food will go up if there's enough money floating around. But you can still get a burrito for $6 if you know where to look.
But the more interesting possibility is actually that we don't increase demand. You have the unemployed, who will go from having no income to having income, and demand from them will certainly rise for the necessities. But for the large middle swath, perhaps this will not add to their spending, but instead provide a means for people to take longer vacations, stay home with children, or create artwork more adventurously. Perhaps this will, rather than increasing demand generally, provide flexibility sufficient for the middle class to feel safer in spending more time in ways that build social ties, benefit the community, and create cultural value that is insufficiently valued in the current economic regime.