The simplest and cheapest way to get to Khao San Road from Suvarnabhumi Airport is to take the Airport Rail Link (ARL) City Line train to Makkasan Station (20 minutes, 35 baht) and then a #556 bus at the bus stop immediately in front of the doors to the station (35 baht).
Don't take an Airport Express train - you'll pay a lot extra to save 5 minutes.
(As of end of 2010, the #556 no longer runs from the Airport Transportation Centre.
This is also the bus you should take in order to get a safe GOVERNMENT LICENCED bus from Southern Bus Terminal to destinations like Samui or Phuket.)
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The answer to your question will depend on which Bangkok airport you are referring to, as there are two of them.
The primary airport is Suvarnabhumi Airport (BKK), which is often referred to as "Bangkok International Airport", and handles all International flights.
The second airport, Don Mueang International Airport (DMK), is the old Bangkok International airport, and although it was shutdown for commercial flights several years ago when Suvarnabhumi opened, it has since been reopened and a number of domestic airlines fly from it.
Over the next few months many more domestic airlines will relocate from Suvarnabhumi to Don Mueang, such make sure you are aware where you are flying to/from.
Most likely you are referring to Suvarnabhumi, so this answer is based on that.
There are a number of options for getting from Suvarnabhumi to Khao San Road.
The best option really depends on the time of day you arrive.
If you arrive any time of the day other than peak period, the best option is normally to just take a taxi.
Including road tolls, the cost will be around 400 Baht (~$13)
through the day, or around 500 Bhat at night, and it will be the fastest option.
All taxis are legally required to use a meter - and make sure that they do as it will be the cheapest option.
In general taxis from the airport will use a meter without you asking, but taxis TO the airport will often try and set a fixed price - if they refuse to use the meter just get another taxi.
If you arrive during peak, or if you're alone and really want to save every last dollar, then there are various other options including buses and train that may be faster than sitting in traffic, but personally I would avoid them and just go with the taxi - especially if you have luggage.
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To get from Bangkok's Suvarnabhumi Airport into town, you could typically take the Airport Express Link or the standard train lines.
Unfortunately, the area where Khao San Road is, is not connected AT ALL by any of their intra-city train lines.
So depending on WHEN you arrive, here are your options (I don't remember that exact prices, I looked these up on Wikitravel's excellent entry on Bangkok ): Take the Airport Express (20 minutes) or slower City Line (15-45 baht) to Phaya Thai (90 baht)
then take bus 15 to Khao San.
This means, however, that you will need to orient yourself and find out where the bus stand is heading towards Khao San and that is NOT going to be easy when you've just landed in Bangkok.
If you give up and try to take a tuk-tuk / taxi, there are more than enough of them around to gouge a high fare out of you.
The cheapest option is to take the free shuttle bus from Suvarnabhumi Airport to the Public Bus Interchange.
From there, take bus 556 to Democracy Monument.
Once you're there, Khao San Road is not that far.
(Seriously, don't believe tuk-tuk drivers who tell you it is.)
Another option at the Public Bus Interchange are private van drivers who pick up backpackers going to Khao San Road.
The schedule for these is pretty much "when it's full".
I don't remember how much I paid (not more than 100 baht, I think, and I don't see any details on Wikitravel for this).
Anyway, these take you either to Democracy Monument or
straight to Khao San Road, and thanks to the stuffy vans it feels a bit like getting kidnapped, but you can't be arsed waiting for the bus you do tend to at least meet other travellers on these.
If your wallet is deep, you could always get a taxi.
How much you get charged will depend on how well you can haggle , so your mileage may vary.
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Take mrt to phyathai then change to bts.then exits sapantaksin station.go to the pier take public boat to pra atrit pier.walk out to main road turn rright 200m turn in small lane on yours left.then
you are there.
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There is lot of way to travel from BKK airport.
Limo : located just outside arrival level door 5 > highly priced about 800-1200 Baht depending on which car do you use.
Taxi :
Easy one: from arrival go to ground level (just downstairs) door
4 or 7 > taxis desk give you a ticket.
The price is meter price + 50 Baht surcharge +
Toll fee is you use Expressway (not needed at night).
Minivan and bus :
Cheapest option.
first you have to go the transportation center >
take the shuttle bus on arrival level door 5
(avoid limo touts) or departure 4 floor door 5.
you should see a blue sign Shuttle Bus.
Bus runs every 20 minutes 24h.
From the transportation center take minivan #551 to victory monument!
(40 Baht) Minivan
#551 to victory monument Note the minivan is NOT convenient if you have large luggage!!!From victory monument you can take bus #12 or 157 (12 Baht) to democracy monument 2 minutes walk to Kao san road.
Sources
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What are my options to go from Bangkok airport to Khao San Road? And what are the expected prices for this?
How to go from Bangkok airport to Khao San?
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<airport-transfer><thailand><bangkok><bkk>
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The simplest and cheapest way would be to take the standard train and/or bus lines, but not the Airport Express train. However, it may be more convenient to take a more expensive taxi or private van taking backpackers to Khao San Road. Limos are available just outside the arrival door, costing about 800-1200 Baht depending on the car. There are a number of means of transport to take you into Bangkok from the airport. Trains are considered a good option, followed by buses. Taxis are the fastest means of transport, especiall during off peak hours. The most expensive option is a limo at 800 to 1200 Baht.
There are trains from Bangkok airport into Bangkok, including the Airport Express train.  Buses are a chepaer option from the airport to Bangkok. Taxis are more expensive but are the fasest option especially at off peak times. A limo can be taken from the arrival area at the airport, costing 800 to 1200 Baht.