Qualifications
Swimming is in your blood. You might as well have fins and gills. If you're a rescue swimmer, you're the bomb.
You'll go to training school with a class of about twelve, and only two or three will graduate. It's that tough. The higher-ups will only pass you if they're sure you can handle the pressure of jumping into the ocean to save someone's life. For real, though, it's a big deal.
Being afraid of heights isn't an option. You'll be jumping out of helicopters often, so get used to it. Actually, being afraid of anything isn't really an option. Except clowns. That's still allowed.
Be rule-oriented. You'll have to learn all the rules completely to fit in and survive—the chain of command, how to address different ranks of people in the military, uniform rules, the no-facial hair rule (for women too)—a vast tome of rules will exist in your head by the end of training. This is the military, if you need to be reminded. You'll have to be exceptional at taking orders and doing what you're told.
Aside from physical strength, stamina, and the ability to swim really, really well, coasties need to be able to think quickly on their feet and make snap judgment calls.
Bravery is required at all times.
You'll also obviously need to pass a physical. Rescue swimmers require special training, including EMT training, and must meet certain PFT standards, like being able to do fifty perfect push-ups and sixty sit-ups.
In addition to being strong, brave, a great swimmer, and a good problem solver, you need to learn to sew. Coast Guard swimmers sew their own gear, repair said gear, and sew hoists together. Finally, all those years of macramé and needlepoint will come in handy.