I'm in the Navy and I work here at the Nuke school in Charleston. There is the old main base, which shut down in the early 90's, and the Weapons Station, which houses the Nuke school and various other small commands. Save for the occasional supply ship (AO or AE), there are no ships here. The Coast Guard has a post near here, but I have no idea what they do.
Interesting factoid--shortly after 9/11, 20/20 wanted to do an expose on the "faulty security of the nation's military bases." They picked Charleston (an easy target, I admit) and tried to hire a boat to get them as close to the Weapon Station as possible. The locals they were trying to hire alerted the military, who sent people out to question them. Oddly enough, this "report" never aired.