Shame I didn't see this any sooner. My only thought is trying to do a constant ping to your ISP's gateway server.
If you don't know how to check what your ISP's gateway server IP address is, try the following:
Go to Start > Run > Type: cmd <enter>
Type:
(Replace "google.com" with the IP of your ISP's gateway server if you do have it)
This will make your computer constantly send a ping to google until you hit <ctrl>+C. Leave this running in the background while you try to get a StarCraft game going, and whenever you get d/c'd again, check your ping logs in this window. If you start getting "Request timed out" errors, than something's freaking out while you're in game. It could be a problem with your NIC, or a number of things. If you have issues with connecting to websites, and at other random times (not JUST when you're on Starcraft), than it could be signs of a problem with your Internet Service Provider.
This isn't going to be as effective as finding what the IP address of your ISP's gateway IP address is, but it's sort of the fool-proof way of testing your internet connection.
You also said you changed your network cables. Do you mean the cables running from your modem to your router? Are you also hardwired from your router to your computer? Or are you running a wireless network from there?
Wireless can be an issue, especially if you have interference in your house.
Additionally, if you're on something like DSL, and on a fringe neighborhood of service, it's likely you have an unstable connection. I had some issues with maintaining constant connectivity until I switched to FiOS.
Hope this helps.