I just remembered that you can buy premethrin concentrate and wash you clothes and stuff in it to impregnate them with the stuff. Some brands of clothes already have this stuff in the clothes. Looks like it's safe for clothes and equipment.
I'd say spray away, especially around the openings. It stays and kills for quite a while. We sprayed our door threshold with premethrin and even after a couple rains we see dead spiders who tried to enter at the door and died.
From a premethrin fabric treatment website: (btw it is VERY toxic to marine life so try and minimize the amount you get in the water):
Sawyer® Products - Water Filtration, Insect Repellent, Sunblocks and More
Q. Will Permethrin ruin my clothes or equipment?
A. No. Permethrin will not damage clothes or equipment. Unlike DEET, which may harm some fabrics and materials, Permethrin is compatible for use even on fragile fabrics such as silk, plus all synthetics and waterproof membrane fabrics. Permethrin will not affect plastics or finishes. IF IN DOUBT, try a sample on an obscure surface area, especially on delicates and check it after 24 hours of exposure.
Sawyer® Permethrin Insect Repellent is odorless, non-greasy and non-staining after it dries. Permethrin can be harmful to aquatic creatures such as fish, so do not spray Permethrin around fish aquariums.