Despite media panic/hype, many airports still have airside mailboxes or even airside post offices. ORD, DEN, PDX, and SEA are good examples.

Airports that don't have mailboxes ususally have extremely paranoid management, idiot FSDs, etc. LAX and SFO are examples. I once asked an airside information desk at SFO where I could find a mailbox, and he told me (in poor English), "All mailboxes are removed. somone could put BOMB in mailbox and blow things up." (said while waving his hands in explosion gesture) If I'd said "bomb" and made that gesture at the same time, I probably would have been arrested. SFO, by the way, is so stupidly paranoid that they moved the mailboxes outside the terminal building. But still on the same side of the street as the terminal building, so little real difference. I guess it's OK to blow up all the curbisde pax but not those at the actual ticket counter.

Of course, the safest place for a mailbox is airside, where all contents deposited in it will have been screened. But some of these idiots don't seem to realize that.