I am assuming you mean the black card the pins come on and not the little black things that hold the pin in place.
Okay, I think that most people do not care much about the black cards when trading. I have not heard of anyone asking for it since they are pretty generic except for the price code. However, if it is a different card that adds to the pin, I keep those. One example is the topiary pins from the Flower and Garden festival that actually have info about the topiary on them or the the ones that have a picture that the pin fits on. That said, I keep some of the black cards around so that when I send a trade I have something to put the pin on. I also throw some out when I get too many. After you really start getting pins, those little things accumulate and just seem to take over, you can't keep them all.
As a side note, I keep the extra backs in a jar since I display a lot of my collection on bullentin boards and don't need the backs, still have to keep them for trading at a later date.