I bought one of those laundry basket on wheels last weekend. I have a half flight down and then a full flight up to get the the laundry - but that is because I like to use the machines in my sewing studio so I don't forget all about them!
Was not a problem until I was going with a load of clean, folded clothes down the steps. About halfway down, the clothing had shifted enough that my clean cloths was now bouncing out of the basket and down the stairway! Ended up carrying it the rest of the way down the steps. I didn't have the lid on it, but not sure if that would be a full solution as all the folded clothes on the top got pretty jumbled up just from bouncing down the steps before it worked it's way forward and started falling out. I only wash my own clothes, and that is less than two regular size loads. The basket was only about half full to start with.
Hmm- maybe I could push it down in front of me instead of pulling it down behind - those that use them on steps, what works for you?