You can get from Bangkok's new Suvarnabhumi International Airport to destinations within the city by express bus. However, express buses serving the city do not stop at the airport. You have to take a shuttle-bus from the airport terminal to the airport bus station -- just go to the bust stop and you won't have to wait more than five minutes. At the bus station, a bus headed for your destination within the city will likely be waiting. A bus tick into the city only costs $1.00. A taxi into town costs ten times as much. In low traffic, the trip takes about forty minutes. In heavy traffic. . . well, I hope you packed a good book to read!
At the airport bus terminal, taking one of the awaiting buses will drop you off very nearby Khao San Road. Just cross the busy street, walk another fifty meters, and you will be there. God help you.
ANOTHER BUS OPTION: I have since discovered that there is another, more expensive bus (150B or $4.00) that runs from the airport terminal to Khao San Road and the river. This bus, which I happened to catch near Khao San Road at the regular "airport bus stop" on my return to Suvarnabhumi International, dropped me right at the terminal. Perhaps it also picks up passengers from the terminal, sparing one the short detour to the airport bus station. . . .
UPDATE: Be weary of taxi drivers and hotel touts as you emerge from the terminal. One of the best things about taking the bus into town is you don't have to deal with such characters at all. Generally the buses -- even the public ones on the airport to city route -- are very comfortable. All the ones I've taken have had air conditioning.
Since you're headed to Bangkok, I suggest you read my post about Bangkok taxis.