Well, pretty much any storage option that is old, handmade, natural materials and/or looks better than a plastic tub.
I aim to buy good quality items in the first place, but occasionally they have accidents or just get used enough that they break down. When a basket starts falling apart, I try to mend it. I automatically turn to fabric for this purpose because I happen to have a ton of it (That might be a literal statement...).
I start with something like this.
I first decide what my color palette will be. In this case, I went neutral.
I snip and tear the fabric into strips - the width depends on the type of basket I'm mending. This one needed 1/2" strips. The fabric was an old curtain. I just use whatever's handy.
Next, I locate my trusty rug needle - of which I have several because they are quite useful and have a tendency to wander off.
From there, it's time to get to mending. This is much like mending clothing by hand. I leave a tail of fabric and stitch over it to hold it firm. Then I weave in and out of the basket, making sure to sew out into areas that aren't falling apart so there's something to anchor the mend. When I get to the end of the fabric strip, I either leave a tail and stitch over it with the next piece, or weave the end out into the basket so it disappears.
It's by no means an invisible fix, but it works for my purposes and it can be done decoratively as well.
Here's one of those ubiquitous wicker baskets. They work great for laundry until the edge starts disintegrating and snagging all of your sweaters. I fixed that with a nice 2" wide strip of fabric.
Once again, my trusty rug needle and some time with Netflix. I just wrapped around the rim trying to cover as many snaggy ends as possible. Do what it takes to "Make it Do."