I will be traveling from Vietnam to Thailand arriving at Don Mueang International Airport.
How can I get from DMK to Sathorn Road and how much would it cost if I take a taxi?
Get in the official taxi stand queue just outside the airport doors and give the person at the counter the address you're going to. They will tell you how much it will cost (or more likely the ride will be metered - it won't matter much either way, as Bangkok taxis are cheap). They will give you a docket, and point you towards the taxi driver who will take you.
Politely ignore any hawkers who try and convince you to go with them in their private car or "taxi". They'll want a higher price. Just go to the official taxi stand queue.
The sensible answer is "taxi", and I hope you have a more exact address because Sathorn Rd is rather long. There are a number of large hotels there, so eg. "Banyan Tree Sathorn" should get the message across, and the taxi dispatchers at the airport will translate English to Thai for the driver.
Personally, though, if I was travelling lightly and looking to save a few bucks, I'd hop on any bus running south underneath the elevated road next to the airport and tell the ticket seller "Chatuchak" as my destination, the fare will be ~20 baht depending on the bus type. Get off at Chatuchak near the Phayonyothin Rd intersection and board the MRT (rot fai tai din), then take it all the way across town to Lumphini station (fare 40 baht). Ta-dah, the northern end of Sathorn Rd, cheaply and without traffic jams. Alternatively, to get to the southern end, go one more stop to Si Lom, change to the Skytrain and go to Surasak.
But I know Bangkok well and speak some Thai, whereas you probably don't, so just take the taxi. ;)
Taxi is always an option, should be around 200 Baht, depending on the traffic. If you arrive during rush hour, it can climb to a much higher price, of course. Here is a quite nice calculator. However, it does not take traffic into consideration.
If you do not mind waiting in the traffic and want a cheaper solution, you can also take the bus No 29, will cost only 18 Baht to the center, but then of course you will have to find and take another taxi to get where you want to, but it stops on sathorn, depends only how far your actual place is from the stop. Check this for the route.
Alternatively, you can also take a train, you will have to walk 500 meters to get to the station. It will be cheaper and you will not have to face the traffic. But it is very slow and does not run often. For the relatively short distance this might be not recommended.
I have used www.bangkokshuttle.com several times to get from the airport (BKK) into downtown Bangkok, but they also serve DMK. Clean, quick, and hassle-free.
DMK to downtown currently costs 1,100 Thai Baht. (Updated 1/1/2017)
Disclaimer: I have no affiliation with Bangkok Shuttle other than as a satisfied customer.