I believe you can download MySQL database from Godaddy hosting control panel using phpMyAdmin, see this http://help.godaddy.com/article/5942. Don't you have access to control panel? Otherwise you can download if your MySQL user is allowed to connect remotely. If you can connect remotely, then you can use any of the desktop MySQL application to download the database.
As the above said, phpMyAdmin is the best option, just export your database. Godaddy is another big host, and they most likely have a custom panel, so ask them if they offer phpMyAdmin.