Hmmm...The one on the right looks like a fruit bushel basket and the one on the left I am not sure but I would guess the intended contents must not be that heavy based on the size of the attached string.
The one on the right is a fruit basket for sure, I actually see them to this day around old orchards that have been in business for a long time. the other one has me stumped . . .but with a handle that "delicate" maybe light garden produce or flowers?
The one on the right is about 20inches high and 16 inches across at the top.
The one on the left is about 16 inches high and 24 inches across the top.
I don't think the hay string was the original handle; if you look you can see hand holds under the rim.