Last time, it my internet was going along with Starcraft 2 just fine, just a little lag when I play WoW, its possible that with both WoW and Starcraft 2 installed at the same time, might cause a disturbance traffic on my connection.
Seriously? I don't think the internet connection could be affected by a program when it's not running. But it's possible your ISP is blocking certain types of traffic or your connection is being overloaded by something else; perhaps even malware. (Malware bytes is a good program to check for those things)