I have a pair of NOS running boards that I would like to bring back to their original luster.
I did a Google on black tire paint and see there are both petroleum based and water based products.
Does anyone have any experience with any of these?
Or any other suggestions.

You might want to start with something mild so as not to attack the rubber. I've had very nice results with Pledge, the furniture polish. Some tire dressings leave nasty residues, not to mention making the surface very slick. Not a good thing on a running board.

I understand Steve's point but you might want to try some mineral spirits to remove the top layer of oxidized rubber. I use it on my tires all the time and it leaves them with a nice matt finish. The mineral spirits will evaporate and not leave any trace. Try it on an old tire first.

It also does a great job on yellowed whitewalls.

You can also sand the top layer, which will remove many imperfections.
For an application, I absolutely love a product called Forever Black for rubber.
http://www.foreverblack.com/products.htm
What I like about it is that it DOES NOT make the rubber shine, but it gives it a very, very rich and deep black without looking wet. One application lasts forever, so detailing becomes a lot easier. Wash with your normal car soap, and the application stays on.

Originally, the running boards had a very thin 'skin that had sheen to it. Over the years they incurred 'shelf where' I filled the nicks with 3M black weatherstrip adhesive. and sanded with 320. I tried a tire black product called No Touch. It works fine on new tires but the running boards seem to have a high carbon content and are porus.
I will visit the foreverblack website. Years ago I had a quart can of a petrolium product that was very thin and reminded me of what came out of the tar truck when I was a kid. Remember the tar trucks and sanders in the city streets?