Currently there are many flights fly from and to Don Mueang Airport (DMK). Unfortunately, the airport does not connect with any rapid transport systems such as BTS Skytrain or MRT Subway.
However, there are bus services running between DMK and Bangkok city center. I have recommended you bus A1 and A2 that runs between DMK and Mor Chit BTS skytrain and Victory Monument respectively here.
Recently (1st May 2017) two new routes are provided:
A3– DMK to Pratunam Lumphini (DMK-Tollway-Dindaeng-Pratunam-Rachaprasong-Rachadamri-Lumpini Park)
A4– DMK to Khaosan Sanamluang (DMK-Tollway-Si Rat Expressway-Larn Luang-Phan Fa-Democracy Monument-Wat Bowonniwet-Khao San-Sanamluang)
The buses are operated from 07.00 hrs to 23.00 hrs everyday with a flat fee of 50 baht per ride.