Given that the location of the USS Enterprise seems to be at least slightly classified (and when they're on fleet maneuvers, very classified), you'd probably need to send your letter in electronic form to Starfleet Headquarters to be forwarded onward the next time the ship makes contact and downloads a message bundle. We see several examples of this in the various Trek shows.
Enterprise
ARCHER: What's the word from home?
TUCKER: The usual, engineering updates. Oh, and Duvall got promoted. They're giving him the Shenandoah.
ARCHER: Duvall got his own command? Thank God we're a hundred light years away.
TUCKER: And I got a letter from Natalie.
TNG
COMM VOICE: Bridge to Captain Picard.
PICARD: Picard here.
COMM VOICE: There's a personal message for you from Earth.