I assume you are doing the same thing as I did for my LED aquarium lighting. I used the arduino to drive a set of 4 Meanwell Eln-60-48p pwm drivers.
Most LED drivers take a 10V pwm signal to power them, (the one's popular in the aquarium trade anyway) the arduino only outputs a 5V pwm signal and you will need a few components to ramp that up. It is a quite simple circuit, and I have schematics of mine, if you would like to see them see them send me an e-mail at email@example.com
But for your problem when you are outputting a 0 to the pwm output on the arduino there should be a constant 0V, which would dim the drivers to 0 regardless of the signal level. I'd assume there is an issue with the code, wiring, or what pin you are using on your arduino.
Could you post your code?