I'm not sure if you're looking for the "canon" answer to this or not.
The canon answer is that it's done by the honor system like it was done in the Age of Sail.
Something I think a lot of Traveller players (and writers) forget is how much the Traveller universe really depends on the "gentlemanly" (honorable) character of people like nobles and starship captains. A lot of Traveller is based on this trust.
In the case of letters you absolutely have to send if you're in the boonies without an offworld mail service (a Postal Union or whatever): You find a reputable captain who is heading to the port in question you want to send your mail. You talk to someone on the ship, preferably the captain (though someone else you trust would work, probably an officer, such as a ship's surgeon or something but not just some crewmember). You give the person your letter and give them a honorarium for his or her trouble. The honorarium was like Cr150 (~$490 2013 US dollars) which is for the trouble. The captain would, upon arrival, make a social call on the party you intend to send the letter to and deliver it by hand. Obviously you probably don't want to send something you don't want someone else reading.
Non-canon: I would imagine for long trips of many jumps and so on, the trick would be to pay the captain to take your letter to a starport with regular mail traffic and the captain would probably write his or her own letter, put your letter into an envelope with the captain's own "cover letter" ("Dear Madam, it's my honor to make my introduction to you, enclosed is a letter from your husband on Catandar's World that he entrusted to me on 210-1049...") and pops into a mailbox. If there isn't anything like that, the captain probably goes to the local captain's club in the highport or downport and finds another captain he or she thinks is reputable who is heading towards the destination, pays that captain a honorarium to carry the letter so and so on. Obviously if you're sending a letter like this, you already know how much it's going to cost for the captain to pass the letter along so you'd increase your honorarium accordingly.
The canon answer is that it's done by the honor system like it was done in the Age of Sail.
Something I think a lot of Traveller players (and writers) forget is how much the Traveller universe really depends on the "gentlemanly" (honorable) character of people like nobles and starship captains. A lot of Traveller is based on this trust.
In the case of letters you absolutely have to send if you're in the boonies without an offworld mail service (a Postal Union or whatever): You find a reputable captain who is heading to the port in question you want to send your mail. You talk to someone on the ship, preferably the captain (though someone else you trust would work, probably an officer, such as a ship's surgeon or something but not just some crewmember). You give the person your letter and give them a honorarium for his or her trouble. The honorarium was like Cr150 (~$490 2013 US dollars) which is for the trouble. The captain would, upon arrival, make a social call on the party you intend to send the letter to and deliver it by hand. Obviously you probably don't want to send something you don't want someone else reading.
Non-canon: I would imagine for long trips of many jumps and so on, the trick would be to pay the captain to take your letter to a starport with regular mail traffic and the captain would probably write his or her own letter, put your letter into an envelope with the captain's own "cover letter" ("Dear Madam, it's my honor to make my introduction to you, enclosed is a letter from your husband on Catandar's World that he entrusted to me on 210-1049...") and pops into a mailbox. If there isn't anything like that, the captain probably goes to the local captain's club in the highport or downport and finds another captain he or she thinks is reputable who is heading towards the destination, pays that captain a honorarium to carry the letter so and so on. Obviously if you're sending a letter like this, you already know how much it's going to cost for the captain to pass the letter along so you'd increase your honorarium accordingly.