I have used a good all purpose cleaner and a brush. The biggest thing is drying the jackets afterwards, fully, and how you store them in the future...away from any moisture.
Have a small fan running in the cabin 24/7. I have my boat on a lift in Clearwater FL and no mildew problems here. My fan runs all year round, keeping the air moving and touch wood no mildew. Still air creates spores, not moving air.