As a native Midwesterner, I was very puzzled to hear my wife (who is from northern New Jersey) use that idiom. I understand what it means, and as far as I can remember I understood what it meant from the beginning; but I thought that my wife's rapid speech was eliding the "to".
My usual sources for etymology (Merriam-Webster.com and Etymonline.com) are designed for word etymology, not phrase etymology, and I don't see (after some search) a question here, so I'll now put the question:
What is the origin and history of the New Jersey idiom "down the shore", used to mean "in the area along the southeastern coast of New Jersey?"
Note The related question, here, correctly identifies the expression as being a local idiom, but doesn't address the question of where the expression came from (For example, was there a local dialect which used "down the" to mean "in the direction of"? If so, by whom was it spoken? Where? Does it survive in any other expressions? Is it at all related to the northern New Jersey accent?) I'm specifically looking for historical origins, which are not addressed in the previous question.