I have successfully used "odor-eaters" insoles containing charcoal to get rid of the musty smell in an old featherweight case. If you do this, you need to seal everything inside a plastic bag so the charcoal has a chance to absorb the odors. I think charcoal briquets or charcoal sold for aquariums would do the trick also, but they present problems if they touch the fabric. The insoles are not a problem and not messy.
While it would be a great idea to air the strips outdoors for a few days, you absolutely do not want to put them in sunlight. Fabrics fade in sunlight.