Salty Dog Blues, Mississippi John Hurt, 1963

"Salty dog, Hey, hey, hey, you salty dog
Oh baby, let me be your salty dog
I don't wanna be your man at all
Baby, I want to be your salty dog

Said, the big fish, little fish swimmin' in the water
Come back here, man, gimme my quarter

Says, God made a woman, he made 'em mighty funny
The lips 'round her mouth, just as sweet as any honey
Hey, hey, hey, you salty dog

Said, the scardest I ever was in my life
Uncle Bud like to caught me kissin' his wife
Hey, hey, you salty dog"

A “salty dog” is an old slang term to refer to a man who spends most of his time at sea (where the salt water permeates his skin, clothes, and hair) and who prioritizes dalliances (girl friends) when he does come to shore.
By singing “let me be your salty dog” and “I don’t want to be your man at all,” the song is asking for a casual sexual relationship in which he’s allowed to spend long periods away and be welcomed with open arms (and legs) when he comes back around.

Note: T-Bone Walker sang "Papa ain't salty no more", meaning of "salty" is here is "pissed" / "upset".

Come here, mama, what's on your mind?
Daddy wants to love you but you won't give him time
You ain't nothin' but a salty dog, oh salty dog
Ha ha ha ha, ha ha, hey, hey, hey

I got a gal, she's six feet tall
Sleeps in the kitchen with her feet in the hall
She ain't nothin' but a salty dog, oh salty dog
Ha ha ha ha

Salty dog = slang for favorite person, best friend or lover.
Derived from the practice of rubbing salt on hunting dogs to keep off ticks. Since salt was commonly in limited supply, one would only do this to their favorite dog, and it became slang for favorite person.

Note
Here's what Dave McKissack from Blacksburg Virginia found: With regard to the term 'salty dog.'
The book 'A Sometimes Westering Man' says that this term actually refers to a medicinal device used by some early frontier America communities, especially in the eastern Appalachian ranges.
In the fall, during hog butchering season, small sausages or "dogs" were placed in a brine solution and left to soak until the winter cold arrived. At the onset of pneumonia or influenza in a community, these sausages, or 'salty dogs' were heated and worn under the clothes, with women placing them inside the bodice or brassiere. It was believed that the combination of brine and hog fat was beneficial because the salt drew noxious vapors from the body while the hog fat insulated the body against cold temperatures (bluegrassmessengers.com).