I have always used a standard plastic laundry basket. But it's a faff to fit under the stairs and always a pain to take it out/put it back.
There is not much space where to put them and they need to stay under the stairs, where the ironing board/iron is (no option to go in any other cupboard). Would it be wrong to buy a fabric laundry basket just for this purpose? They are normally folded up neatly and fully dry....
It would take less space and as it's fabric it would be easier to squash in....