If you have your controller properly set up in Windows, just launch the game. The DOSBox logo will show and, once the game is launched, press Ctrl+F1.
This will open the Mapper tool. You can use it to assign controller buttons and axes to kayboard buttons. The game will keep using keystrokes for movement, but you can use your controller to trigger them.
Just click the in-game button (like the UP keyboard arrow), click the "Add" button, and use the appropriate function on your controller (for instance, move your left stick up, or push your D-Pad UP button).
Once you're done, click Save. You should now be able to use your controller in-game.