Actually, they're not set to up and down arrow at the moment - that's what the digital controls are mapped to, which are just extra keymappings so you can use keys/buttons to play the game (badly) if you have no analog controls configured. The trackball settings you need to fix are actually called 'Track X Analog' and 'Track Y Analog' in that pic and should be mapped to the mouse axes.
If you do need to configure them manually, select Track X Analog, hit enter, spin the trackball left to right, and that should be it (similarly for the Y axis).
But, you shouldn't really have to go through and configure each game this way on a case by case basis - if mouse input is enabled, then the default setting is for analog X axis and analog Y axis to map to mouse X and Y anyway. The fact that it isn't like that on centipede (or whatever's shown above) probably means you've gone in and dicked about with the configuration for that game before you enabled mouse input, so now you have a saved config file for centipede that isn't set up to include the mouse - go and delete all your configuration files and they should all revert back to the defaults.
You'll probably have to adjust the sensitivity of those axes to play the game properly, which is more of a laborious game-by-game process since different games used different sized trackballs with different optics etc, no one setting is going to give the right feel for everything.