I'm actually running a game right now with a few characters who remember the pre-apocalypse (they were kids). It was actually important for one character because, in effect, he had some rudimentary farming and had been creating a sort of utopian commune where people could actually produce food (food scarcity being one of the themes of our game). The other person who can remember the pre-apocalypse is treated as a sort of sage by all the power players in the area.
I think the nice thing about having characters who remember the pre-apocalypse is that I think, ultimately post-apocalyptic stories can deal with interesting themes about death, loss, destruction, and rebirth, and having someone who has some kind of way of highlighting that, albeit in a vague way, gives you some narrative hook to tag that theme in play.
Basically, if no one knows that something was lost, something that used to be, how can you explore that theme?
Personally, as far as use in the game, I've tried to keep the two NPCs who have any kind of pre-apocalyptic knowledge off camera a fair amount and display most of their knowledge subtly. I think the danger with this is of having an old guy dropping into the fiction to monologue things like "And back then, we used to have Coca-Cola, and the Internet, a jet airliners, and movies! You kids don't know what you're missing!"
I find it's least intrusive and most effective to have the knowledge manifest not in what the characters say but what they know. That piece of trash lying around? The character can do something with it, or knows what it is. That sort of thing. One of the NPCs in my game who remembers the pre-apocalypse actually has plants and watered them. This looked real strange all around to the PCs. Water is scarce, and green stuff even more so, why waste water on some weird little green thing? That sort of thing.