You can use a USB ethernet adapter and CAT5 patch cable for connecting your Wii with wired Internet.

Airlink ASOHOUSB is a USB2.0 Ethernet LAN adapter. It allows to create a high speed Internet connection for your Nintendo Wii game console or Windows PC. Just plug the adapter to the USB port on Nintendo Wii or your computer, then connect it to your Cable/DSL modem (or router) using CAT5 patch cable. Now, you are ready to enjoy your broadband Internet connection.