There are showers at Jetty Park which is the closest beach to Port Canaveral.......also a nice fishing pier to stroll and check out the catch of the day.
Jetty Park is right at Port Canaveral. If you have transportation there's a nicer beach farther south on A1A called Lori Wilson park.......also has showers and a nice nature trail.
Cocoa Beach is a nice beach that is not too far. There is also the world famous Ron Jon Surf Shop there.
If you want to go to a really beautiful beach, then i would suggest the Canaveral National Seashore. Its a few miles north of the port, not sure if there are showers/bathrooms there or not. Its a national park and very pretty and not crowded at all.