How do I re-make a basket with good bones?

I have storage cube that came with baskets. The bones are make of wood strips so very solid but the wicker on them is broken and brittle. I want to re-make the sides of the baskets (there are 5... two big and 3 little) and re-use them elsewhere as the cube is no longer good. I want them to be sturdy for storage of items and I do not like the twine or rope as my cat will try to eat string and twine. I like burlap and canvas but how do I inexpensively make the sides sturdy under the burlap? This is my first diy project so need something kinda simple. I am still learning.
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