Datth on the forums confirms that WoW is causing some home networks to go down completely. After hearing reports that on some networks, logging into or playing the game would cause the entire network to lose their internet connection, Blizzard asked players to report technical specs if the problem happened. And after researching, they've reported now that the problem supposedly lies with folks who have a "Siemens, 2wire or older model Motorola modem or Netgear router."
So if you have one of those and are having this problem, Blizzard has at least identified the cause. They say they're working with the router manufacturers to come up with a fix, but if there really is a problem in the router itself, odds are Blizzard won't be able to hotfix it on their site, so you may have to go to the router's manufacturer for a fix (all that information, when it becomes available, will likely be on their site).
I am surprised, however, that this problem didn't appear before now. Blizzard says that the most recent patch (would that be 2.3.3 and the improvements to load time?) is responsible for "revealing a bug in the hardware," so if the router manufacturers don't release a fix, Blizzard may have to reverse whatever changes were made. At any rate, a fix is coming, no matter where it comes from.