The city bus "S1" or similar is cheap. It leaves directly from the airport. The bus is already waiting for you 10 steps out of the door. The bus leaves every half hour, from 08:00 to 23:00.
Duration: 45 minutes
Price approx. 2€ to 6€ /60 Baht
Provider 1 : Bangkok Mass Transit Authority S1
Provider 2 : Airport LimoBus Express Suvarnabhumi Airport - KhaoSan Road
Duration: 25 minutes to 45 minutes (without traffic jam)
Just ask for a bus to KhaoSan Road in the airport, just before the exits to the outside. Most likely level 1. There should also be stalls directly visible and signs with bus + KhaoSan Road.

The unbeatable advantage of the tram in Bangkok is always the congestion-free journey and the very reasonable price.
The tram in Bangkok goes through the most popular areas in Bangkok. Unfortunately, it does not go directly to KhaoSan Road. The nearest station would be the "Phaya Thai BTS Station". This is 5.4 kilometres away from KhaoSan Road. Solution would be to get off at this station and take the bus. by train
Duration: 26 minutes by train
Airport Rail Link - City Line
Price: 1€ - 2€
Frequency: Every 15 minutes
+ bus
Bangkok Mass Transit Authority
Duration: 11 min.
Price: 1€
Frequency: Every 10 minutes

The taxi ranks are located directly at the airport in the lower area of Terminal 1. Just wait in line and you will be assigned a taxi. Costs about 400 baht (9 to 12€). Just ask other people and share a taxi. You will also be offered rides by individual taxi drivers for 1000 baht or more. Friendly say no and move on. These drivers will take advantage of your long journey and many just want to get to their hotel. Ignore this and take an official taxi.

Remember the 400 baht, because if you ask, you will get a slightly higher price (500 baht).

Depending on where the hotel is, a taxi from Bangkok to Khao San Road is necessary, as the bus may not stop directly at Khao San Road. So the regular bus is a bit more complicated but much cheaper.

By Syktrain from Bangkok Airport to Khao San Road
The fastest option as it is independent of traffic is the Skytrain (Suvarnabhumi Airport Rail Link).
Be sure to take the "Express Line". The train goes directly to the city centre in only 16 minutes. So its non-stop to downtown Bangkok.

You get off at the Phaya Thai terminus. This offers further transfer possibilities, for example to get to the popular Sukhimvit Road (Nana nightlife). It is relatively easy and for an onward journey you would simply take the "Sukhumvit" line.

Taking the Syktrain from Bangkok Airport to Sukhumvit Road

If you have a hotel on Sukhumvit, take the Airport City Line.
The line stops at several stations and therefore takes about twice as long as the Express Line.

Bangkok Suvarnabhumi Airport
Stop
Lat Krabang (City Line only) Stop
Ban Thap Chang (City Line only)Stop
Hua Mak (City Line only)Stop
Ramkhamhaeng (City Line only)Railway Station, Station
Makkasan City Air TerminalStop
Ratchaprarop (City Line only)Terminal station End of line
Phaya Thai

Travel to Phaya Thai Station and change to the Onnut-Bearing Sukhumvit Line towards Bearing. 5 stations further you are at Nana Plaza Station (E3). This is also where most of the hotels and the biggest nightlife in the area are.