Sounds like a faulty accelerometer. It could possibly just be stuck, or it could be completely broken.
The only fix you could try yourself is to give the remote a whack to try to unstick it.
From here: http://www.nintendo.com/consumer/sys.../wiiremote.jspQuick tip: Cursor (hand) or character movement not responding properly?
Hold the Wii Remote with the button side down. Without using much force, firmly tap the button side of the remote against the palm of your other hand a few times. Doing this does not take a lot of force, so be sure not to tap the Wii Remote very hard. Check for proper response. If the problem is not solved, please follow the rest of the instructions on this page.
If that doesn't fix it, then you can contact Nintendo, and maybe it might be in warranty still and they will repair/replace it for you if you send it to them, or you'll just have to buy another. I don't think it would be worthwhile to pay Nintendo to repair it if it is out of warranty considering the price of a new one.