I feel your pain!!
I'm dealing with this right now with my 1991;
as a matter of fact, my new (used) ABS assy. is on the way this week.
A few weeks ago I did an EXTENSIVE search b/c of the same issue and didn't really see it explained as well as I'm about to do, so here goes:
As I've been told by Nick E. at Applied Motorsports in Oceanside (NSX ONLY mechanic):
Constant pumping (start/stop) is due to a sticking solenoid inside the ABS system. The pumping noise you hear is, believe it or not, the ABS air pressure pump working perfectly!
It's attempting to fill the ABS sytem with air pressure, but because of the sticking solenoid, the system sensor thinks that the system has only a tiny bit of pressure, but not zero pressure (see end of my reply for why that's important). The pump goes until the sensor just BARELY senses pressure, and stops. But just then, it senses BARELY too little pressure, and it pumps again!! And so on, and so on...
This is generally caused by UNDERUSE of the ABS system over the course of it's life. IOW, the NSX's ABS system is meant to be used frequently, and can get "rusty", so to speak. Your NSX has been babied too much!!
Now the bad news...
Unfortunately, the NSX's ABS system is a closed system, meaning you can't open it up and really work on/replace that solenoid. Bottom line: if you can't get it working again, you have to replace the entire unit, cost new: >$2,200, not including labor
How do you get it working again? Only 2 things you can do:
1) Have your mechanic THOROUGHLY flush your ABS system. There's a SLIM chance there may be a small piece of crud in the system causing the solenoid to stick, and that may work.
2) Excercise the system. You need to go out in some rain or sand (SAFELY, of course) and lock 'em up. A lot. Many times. For a while. And pray that the solenoid un-sticks.
If that doesn't work, than you are unequivocally SOL (As I am!!)
It took me 5 months to find a reasonably priced, used, newer unit by itself (The WHOLE UNIT, not just the modulator). You may find one quicker, but expect to pay no less than $500 for a decent unit. Again, new you're looking at >$2,200. Acura list is around $2,400.
What can you do in the meantime to quiet that embarrassing noise?
Remember, the sensor thinks that there's a LITTLE pressure in the system, but not zero. If it were zero, the pump would pump until it gives up, thinking that there's a system problem. You need to create zero pressure so that after 2 minutes, the pump will at least shut off. The ABS light will go on, but that's fine. I know how you feel, you just want it to stop PERIOD.
The ABS sensor harness is located just to the right of the spare tire and down. You'll have to squeeze your hand in there, but the sensor unplugs if you pull up on it after moving the securing clip to the plug.
I'm sorry I can't give a better description; ask your mech where it is, or if he can just unplug it. You can leave it resting on the connector, but not plugged. It will hang out there while you drive.
This will cause the pump to start pumping upon drive off, and pump constantly for 2-3 mins. Then, when the system recognizes zero pressure, the ABS light will come on, and the pump will stop. This will repeat every time you turn the car off/on, but at least after 2-3 mins while driving, the pump will not pump, and you can drive around without people looking at you.
NOTE: YOU WILL NOT HAVE ABS AT ALL IF YOU DO THIS. BE CAREFUL
Good luck!!