I had to dig for the information but it looks like there’s at least a mail drop and a good place to buy stamps.
For the South Terminal, there’s a pre-security mail drop on the departures level by the Starbucks and the Alaska Airlines e-ticket check-in terminals. Note that the mail drop might be outside; take the necessary precautions if it’s cold outside.
The FAQ also states that there’s a drop box for the North Terminal “just inside the North entry doors”. No map is provided; the link is broken.
The FAQ indicates that Alaska Luggage and Seafood Shipping, in the South Terminal level 2 (departures) across from the United/Delta check-in counters, has stamps for purchase. This is on the other side of the elevators relative to the South Terminal mail drop.
Source (you need to click “Passenger Services – FAQ” for the latest FAQ)
Last updated 17 June 2017