From late Middle English, a noun or adjective. It evolved like this:

1. (obsolete) (only in Middle English) a board used to cover a porthole.

2. (obsolete) (only in late 15th - middle 17th centuries) the seaward side, esp. of a ship. Used only after prepositions, e.g., at seaboard, on seaboard, to seaboard.

3. (obsolete) (only in middle to late 16th century) the sea as used by ships. Only as "by seaboard", by sea, on a ship; "on seaboard", at sea, on board ship.

4. (from late 18th century) the line where the land and sea meet, the coastline; the seashore or the land near the sea, especially considered with reference to its extent or configuration.

5. (from late 16th century) (adjectival only) situated on the seaboard, bordering on or adjoining the sea.

[adapted from The New Shorter Oxford English Dictionary, vol. 2. Oxford, UK: Clarendon Press, 1993]

The Maven's Word of the Day has this to say, which is not especially enlightening, but gives me a clue.

Note the dates when "seaboard" became used to describe geography rather than ships. This was the heyday of British America, where the Eastern Seaboard is located. It could be that the Eastern Seaboard refers to the East Coast of the United States because, during the time when "seaboard" was in more common usage, the only seaboard in America was to the east. It's directional, like West 53rd Street or North Market Street.

We don't have a Western Seaboard because we had dropped the word as a common descriptor by the time we got to the West Coast.