From what I recall seeing: there is a mail slot at LGA ground side main terminal that I think is part of an actual post office. I also remember seeing mailboxes at SFO outside the front doors at the terminal that UA uses. I don't remember seeing any mailboxes airside though.

FWIW I was able to mail an AMEX bill once by asking a favor of the matron at the UA International First class lounge at SFO - I showed my 1P card to show that I wasn't totally unwashed though.