It's all fun and games until someone loses some cashmere. First thing, spray down the closet with a non-staining insecticide like SLA. Secondly, don't store dirty clothes. That's not to say you're dirty, but rather make sure you wash or dry clean any pieces before putting them away. Moths (and other fabric munching bugs) are drawn to food stains, perspiration and any other trapped moisture. When you drop off your clothes at the cleaners, ask for them to be folded instead of hung—making it easier to box up. Cedar, a time-tested deterrent, not only keeps pests away but also absorbs any moisture while you've got coats and sweaters tucked away for the season. Rubbermaid makes handsome and handy under-the-bed canvas boxes. And of course, you could always try Space Bags.
Moths don't eat cotton. And adult moths don't eat fabric at all. Larvae feed on fibers.