We air-dry our laundry on a foldout drying rack in our bedroom. During summer, we take that rack outside, but there's often some heavy wind on our balcony, and clothespins are necessary to prevent the clothes from getting blown away.
We used to have a cute vintage clothespin bag, you know the ones that look like a little dress, but sadly we tossed it when we moved into this apartment, and use a simple cardboard box instead to store the pins. However, the box is really impractical.