When you say "Silom line", do you mean the BTS (skytrain) line that runs from National Stadium to Wongwien Yai? I think the answer is that there is no direct transfer.
The website http://www.bangkokairporttrain.com/index.html says
- * - * -
The Airport Link to Suvarnabhumi Airport will provide 2 Transit options:
* A Direct Express Train that will take 15 minutes from the City Air Terminal (Makkasan) train station. This service will connect to the city