If you live in a moist climate or it’s a moist season, hanging the herbs is likely a better approach than the basket.
For this method, loop a string around the herb stems and hang the bundle high because heat rises. I have a baseboard in my kitchen with little nails along it, and I hang the bundles from that. As the herbs hang, make sure they are exposed to air all the way around. Again, you don’t want them in direct sunlight.
If you’re dealing with a more moist plant like peppermint, you’ll want to hang individual stems. If a tiny bit of the leaf folds over on itself, the place where it folds can turn black, and you don’t want that.