I had no problem backing up The Day After Tomorrow region 1.
Obvious question to you but start with the basics.
Make sure you have AnyDVD running before you insert the original DVD.
If you see the Red Fox on your task bar, it is running. Click on the fox and make sure under the DVD tab, and feature removal section, all boxes are checked.
If it is I would uninstall AnyDVD and then re-install and see if that corrects the problem. Let us know how you make out.