Required? Very few. Even Windows would happily run without a mouse.
Benefitted from: A number of DOS paint programs were a lot easier to use with a mouse. Indeed, PCPaint and PC Paintbrush generally were shipped as part of retail mouse boxes. Logitech didn't include a paint program but instead included the Point text editor which exposed many of its features to a mouse. MS Word was one of the earlier DOS word processors to support a mouse. Don't forget the other GUIs like Visi*On and GEM.
Games: Flight Simulator. Battle Chess. Balance of Power for all that it was a Windows game was difficult to use with just the keyboard.