The Old Mariner can be found on the eastern part of the beach, which must be unlocked by repairing the small bridge. He hangs out there only in the rain (he is not there when it is snowing).
If you get a house upgrade and are at a ten heart friendship level with one of the marriage candidates, he sells you a Mermaid Pendant (5,000 gold) , which you need if you are planning to marry someone.