This is a really good question. For a while while I was pursuing 1.1 million in DK, I was using a PS3 controller. During that time, I experimented with several different kinds of pads - NES (with NES to USB conversion), iBuffalo Classic (SNES-type controller), PS1, PS2 and PS4 controllers, the 8BitDo M30 and Hori's Pokken Tournament DX controller. I even tried the Mayflash F300 Fight Stick.
The most difficult part of using a controller for DK was trying to make sure you never accidentally pressed in a direction you didn't want to go (i.e., when moving right or left over a ladder and you accidentally press up to go up it instead of through it). For reasons like this, I found any of the PS controllers to be, by far, the best. The PS3 won out for me due to its comfort over the originals. The PS4 has up/down/left/right integrity issues, so I can't use that. For all of the easy-to-open controllers (NES, iBuffalo, 8BitDo M30), I even went as far as to add black tape to the insides of each direction on the dpad circuit to prevent errant directionals. This helped but still, nothing beat the PS3 controller, which I actually ended up getting my first million on.
I will say that throughout my whole journey, my left and right thumbs began to ache terribly. I started putting all kinds of creams/ointments on my hand joints practically every night just so I could grind the next day. For this reason, I would say a keyboard is probably the best fit for MAME DK. I got myself a Hyper X Alloy RGB mechanical keyboard and I haven't looked back. But to each their own!