So it's been a long damn time since I've posted in the Pop'n forum, and mainly that's because I've had trouble actually finding the time to play Pop'n. Even though free time was abundant for a while, it was never a good time to play. Apartment living and a loud as hell controller can do that to a man.
However, a few trips to 8 on the Break later and a renewed addiction to Pop'n forming, I figured it was time to bring an end to this problem, and after some experimenting, I've found a pretty good solution. It doesn't completely eliminate the noise, and the controller will still be decently loud, but it will reduce it from an earshattering crack to a much more respectable low thud. Another added benefit: If you don't like the results, the process is cheap and easily reversible!
(NOTE: I did this with a Ransai, but the same general idea should apply to a DJ Dao, a DASC, a Controllica, a homemade ASC, a KASC, an actual cabinet, etc, etc.)
There were a few different materials I tried. Pipe insulation was too hard to cut thin enough, felt was about right but too thick and made the buttons not register too often, and hobby foam was too solid to actually allow any airflow on the underside of the button, so it didn't compress very well at all.
The winner was, oddly enough, a plain old wifebeater (or an A-shirt to be politically correct) that I chopped up.
So! Here's what you'll need to complete this whole fun process:
- Your Pop'n ASC (durrrr)
- The shirt
- Scissors
- A Sharpie or other permanent marker
- Scotch Tape (not the clear kind, the cheap crappy stuff is better for this)
The first step you want to take is to remove your buttons. I'll just be doing one for the guide, but the same process goes for all nine buttons. Pop the switch out, unscrew the nut holding the button in place, and bam! Button.
Now on the underside of the button, you should see two small white tabs. You can pinch these inward and then press them down, and this should separate the white and colored plastic pieces from the black plastic holder.
It is at this point that my camera died and I switched to my phone. Sorry for the shitty picture quality from this point onward. 
Now this next step will help you make sure you're cutting the shirt up into the right sized pieces. If you look at the side of the white/colored part of the button, you should see how they stay together. It's really just a matter of gently prying the colored cap off the rest of the button, and then you'll get a white base, a white separator, and a colored dome.
Take the middle piece (the flat white separator) and lay it out on the shirt, and trace around it with your handy dandy Sharpie. If you're worried about getting marker on the separator, trace it with pencil on paper first and cut that out to use it as a template. Once you draw the circle, cut it out of the material a decent bit smaller than you've drawn it. You want the circle to be a decent bit smaller than the actual separator you used as a template. (After you've used the separator as a template piece, you can simply put the white pieces and the dome back together the same way you took them apart.)
Once the circle(s) are cut out, fold them over twice, and cut the middle corner out, so when you unfold it there's a hole in the center. If you place it on the underside of the dome part of the button and it looks somewhat like the result I have here, you should be OK. If it's hanging off the edge a bit, trim it, otherwise the button could get stuck once you've finished.
Reassemble the button completely, and install it back in the controller once you're done.
Once the controller is reassembled, TEST IT. If it works and feels fine with no misfiring, congratulations! You're done! Don't overcomplicate things with the next step if you don't need to. Mine was almost perfect at this stage, and I only needed the next step to prevent the occasional dead hit, which was about four or five per song while playing 40s and 41s.
On to part 2: What to do if it still isn't perfect.
How to: Make Your Pop'n ASC Quieter: Without losing sensitivity! (Lots of pictures) 
