Im currently using a USB to Ethernet adapter connected from my non-wireless linksys router to my Wii. The connection is fantastic and fast and this adapter works first time without having to install any software or codecs or run a cd or anything, just plug and set up the connetion in the wii settings.
Heres the newegg product link, its only around $20.00 usually, or it was when i purchased it, and its usually in stock.
KINAMAX USB 2.0 A-Male to A-Male 3ft Extension Cable Model USB-AMAM - Retail
This works, there are videos on YouTube of it working, and there is another Wii message board on which dozens of users including myself have confirmed it working. Im posting this because i noticed it wasnt stickied and it should be and i saw an earlier post about people wondering if there was something that worked in the USB port.
Heres the link to that youtube video.
Unofficial USB to Ethernet Adapter Working on the Wii
Heres that original post on the nintendo forums.
Working USB To Ethernet Adapter for the Wii
Well i hope this isnt a repost or unwanted but i didnt see anything about it on this forum so here.