This is a question that has an intricate answer in my case. I can't really speak about Donkey Kong, but I definitely can on Street Fighter 2, Mortal Kombat, Pac-Man, Do Run Run and Contra.
My first play on Street Fighter 2 was in an arcade, and when I was learning the motions for the specials I found it very frustrating on the joystick, (especially the dragon punch). It was in-fact so frustrating that I stuck with charge characters like Blanka and Guile (who became my mains). When I picked up SF2 at home on my SNES, I was actually surprised by how much easier the motions were for me. Ken became one of my better characters.
Now oddly enough, Mortal Kombat is the exact opposite for me. I have a much harder time on a pad than with an actual arcade control panel. For me it has more to do with being able to quickly tap two buttons simultaneously; I can do that easier with my fingers than trying to position my thumb over oddly positioned gamepad buttons.
Pac-Man has always felt wrong on a directional pad, and I just can't get used to it without a joystick. Period.
Ditto with Do Run Run, needs a proper arcade panel to play well.
Now Contra is the odd one here. Having played both home and arcade versions, and also seeing how they are both different, I can play each one well with its respective controls.
The arcade Contra moves at a slightly slower pace than on the NES, and I find that the slower speed helps me perform better with the joystick there, getting my positioning right for the diagonal shots, and standing in the "safe spot" during boss battles, as any Contra fan knows well.
Wow. I better stop here before I write an entire book.