The Laundry Baskets are used by Prisoners and Janitors to carry dirty uniforms to the Laundry Room and clean uniforms to Cell blocks. Prison Labour and Janitors each require their own research in Bureaucracy.
Make sure not to purchase more than 1-2 baskets in a new Laundry, as having too many will clog up the system. Prisoners will only deliver clothes once a basket is full and the current AI (A 10) spreads the clothes out in different baskets.
Don't think baskets wait to be filled anymore they seem to cycle every day regardless. Most prisons will want one basket for each cell block, if the cell block contains 16 cells.