I love getting fresh herbs in my CSA. They’re always bigger and more impressive than the herbs I grow and, because they’re freshly cut, they inspire me to use them up quickly. Depending on how hot the weather is, most soft-leafed herbs can do fine in a cup of water sitting out on the counter for a few days, but rosemary does better in the refrigerator.
*My favorite towels for food storage are flour sack towels, which are inexpensive, durable, and thin enough to be used like a paper towel. They come in packs, and I recommend keeping some full-size for vegetables like kale and cutting some into quarters and half sizes for herbs or other small veggies.