wii supports WPA2-PSK TKIP along most others, if you wii cant connect to your router when the security is on then you have an issue with the SSID or network key or both or the access control list ACL for short, ACL resides in the router and is like a list of devices which can connect, when there is no security there is no ACL, and when the security is up then it refers to the ACL, there is one other aspect aswell, most routers can be set to auto accept new connections, or by register only, or by not accepting connections, you will want to check them too.
If after you checked everything then really the only thing can be at fault is the router, but dont despair as you can reset the router back to factory settings, and that will delete a little file in the router storage memory called a config file.
I know with some D-Links you have to save the changes and they are in some weird ass place
I have mine connecting fine on WPA2-PSK TKIP fine
also make sure connect on demand is not ticked that way your router wont drop the connection