Yes, the rents in popular areas will rise. Since people have a basic income to support them regardless where they are living, they can move to areas with lower rent (thus increasing the rent — and net rental income — of the area they move to).

At present, most unemployment benefits systems require the recipient to stay where they are, or move to places with better employment opportunities. This places a large burden on the recipient since they might already be in a high rent area and are not allowed to move to somewhere they can afford to live.