Storage baskets used to mean one thing: a sad plastic bin under the bathroom sink, or those wire mesh baskets everyone had in the early 2000s that rusted the second a wet towel touched them. Now there's actual range, woven seagrass, water hyacinth, cotton rope, proper rattan with reinforced bases, and most of it is better made than it needs to be for what it's asked to do. That's a win. The only downside is you can now spend quite a lot of money on something a laundry hamper would have done for a tenth of the price, so it pays to know what you're looking at before you hand over cash for a basket that's basically a tote bag with ambitions.
Flip one over before you buy it, same as I'd do with any piece of furniture, because a basket lives or dies on its base and its handles. A well made one has a flat, reinforced bottom, sometimes with a wood or metal ring sewn in, so it doesn't bow out and spread when you load it with blankets. The weave should be tight and even, no gaps you could lose a sock through, and the handles should be stitched through the body of the basket, not just glued or tacked to the rim, because that's the first thing to fail and it fails fast. Natural fiber baskets age well if you keep them somewhere with a bit of airflow, seagrass and water hyacinth don't love constant damp, so a laundry basket in a steamy bathroom will eventually go brittle and start shedding fibers at the rim. A light vacuum now and then and keeping them out of direct sun, which dries and cracks the fiber, is most of the care they need. Cotton rope baskets are more forgiving, you can even spot clean them, but check whether the rope is wrapped around a wire core, because that's what gives it the shape to stand up on its own instead of slumping like a tired sock.
Measuring up matters more than people expect. A basket that looks generous in a shop photo often turns out small once it's next to a sofa arm or under a console table, so measure the actual gap, not the vibe of the gap, and go a size up if you're torn, since an oversized basket reads intentional and an undersized one just looks like you ran out of room for your stuff. Check the day it arrives that the weave isn't crushed flat on one side from shipping and that it sits level on the floor without rocking. I'd start with one in the living room, under a side table, holding the blankets that otherwise live in a heap on the armchair.