^^ You guy's probably don't understand the question. It seems that not many people on this forum know about the USB adapter (it plugs into your computer to enable your DS or Wii to connect if you don't have a wireless router, your computer has to be on because there is a software app that has to run for it to work).
The answer is: you don't. This would be just like asking if you can check your email without your modem connected. Although, there may be a feature where you can download anything you missed while you were diconnected (maybe?)
OK, here's how this works... If you don't have a WiFi router, and don't want to spend the money on one you buy this USB key from Nintendo. This does not plug into the Wii, it goes to your computer. When you set the prefrences of your DS or Wii to connect to a USB connector, you username shows up on the computer, you must confirm it (for secruity) the first time, after that a message just pops up that a user has connected. In order for the USB key to work and get a connection on it, the software has to be running. The only way for the software to run is for the computer to be on.
note: for the USB key to work, you must have a wired broadband (everything but dialup) connection on the computer you are connecting it to.
I would think that you could catch up on downloads when you start up in the morining, but anything where others connect directly to you (like the Animal Crossing scenario) will be lost, since those features would require others to directly connect to you. It would be just like you unplugged your Wii. This could be solved by using a WiFi router that stays on all the time.