Origin of New Jersey idiom "down the shore"
It appears the use of down is taken from local dialects and means; to.
down the shore,that is, to the shore
” In New Jersey, you invariably go “down the shore.” Baltimore natives, meanwhile, say they’re going “down the ocean” — but in Baltimorese (make that Bawlmerese), the phrase sounds more like “downy eaushin.” The down of “down the shore” and “down the ocean” doesn’t necessarily imply a southward journey. As in many dialects along the Eastern Seaboard, down can be used as a preposition indicating movement from the inland toward the shoreline.
Source: www.nytimes.com
I've bundled my comments up into an answer in case the comment monster gets them at a later date.
A local dialect like BrE?
In common use in British English is the phrase
I'm going down the pub
which means
I'm going to the pub regardless of what compass bearing I have to take
The same type of phrase can be used in going down the town, up the snooker hall, down the club.
Matt Gutting added:
Exactly. "The Shore" is roughly a strip of land (and the included municipalities) from halfway down the east coast of New Jersey to Cape May at the extreme southeast corner. Going "down the shore" can be done from anywhere within a hundred miles or so of the shore, and simply means "going to a destination in this area".
That's pretty standard BrE.
We go down the whatever or up the whatever everywhere.
You can go up the town for the sole purpose of going down the pub.
You only need a degree of directionality if you are going somewhere distant like going up North, or down to Brighton.
Anything in an east or west direction doesn't have the distinction (the UK is not very wide) although if you were going to the west end of London you would say I'm going up West, even if you have to travel south and east to get there. The east end is almost always down, down the East end no matter where from.
It also does not require that you are intending to, or actually are, travelling. If you get a phone call and someone asks Where are you? you can reply with I'm down the pub.
To show just how common this is in BrE here's punk rock band Sham 69's "Hurry up Harry" from 1978, which makes considerable use of the phrase We're all going down the pub. http://www.youtube.com/watch?v=LdVFzdRZF0Q (suitable for work but rather loud)
Going "down the shore" means driving from Northern NJ to southern seashore destinations. Pretty simple. To "drive down the shore" means to drive south on the Parkway, to the seashore.