I have a tradition between me and a few friends to always send postcards from airports I go/transit through. I get all kind of cases:
- mailbox airside: just put it in, as long as you can buy a postcard (and if necessary stamps) from airside shops (never failed me)
- mailbox only landside: fine if it's your originating or destination airport, otherwise you have to consider the time and hassle of exiting and going through security again (and if you have duty-free liquids from a previous airport in your carry-on, you can't.. yes I did that once)
- mailbox nowhere in or around the airport: that is extremely annoying, the only case I know of is BUF

If I can't find a mailbox, I usually try to ask the shop where I bought the postcards or the SkyClub (nicely). Shops also always know where the mailboxes are.