This is how I got my Wii to work with an older, non-Extreme Airport Base Station:
My somewhat relevant Airport Base Station settings:
* Wireless Security is 40 bit WEP (128 may still work)
* Create a closed network box is checked
* MAC address based access control is turned OFF.
0) If you've changed the Wii's internet connection settings, please clear the settings first before going to step 1.
1) Open Airport Admin Utility, and click the name of your base station
2) Click Configure, then enter the password to open its configuration panel
3) Click the padlock icon, "Password." This is the important step! This opens a dialog box that tells you the WEP key to use for any WiFi card that is not an Airport card. Write this down.
4) Go back to your Wii internet "Connection Settings". If you do not have a closed network, you can have the Wii "Search for an Access Point". You can then select the network from the list. If you do have a closed network, then select "Manual Setup" instead, and enter the name of your Airport Base Station network into the "SSID" field.
5) Once you have selected or entered the network name, you should be able to find the "Type of Security" page. Click on WEP. Now enter the WEP key you wrote down in step 3. There should be a Confirm button to confirm the password. Click it to make sure that it worked. Leave all other settings at their defaults.
6) You should be at the screen saying "Connection settings have been saved...[blah blah blah]" Click OK to have it test the connection. This will take a little time. If it succeeds, you're done! Yay!
See page 56 and 57 of of the gray Wii Operations Manual for more details on the internet configuration settings and how to get to each screen.
If you run into any problems, make sure that:
* You copied the WEP key correctly
* You entered it on the Wii correctly
* You typed the network name (aka SSID) correctly
* Your encryption type matches what's on the Airport Base Station (WEP, WPA, etc).
* If your Base Station is configured with Access Control - one or more MAC addresses are listed in the Access Control pane - then you will need to locate your Wii MAC address from the Wii settings (that info is elsewhere in the forums) and enter it into the Base Station, then update the Base Station. Then try the above steps again.