I don't have kids, but my fellow Trekkie
@StefanM does. He is currently not online, but he might share some stories. I know that he reads Trek books with his children.
Yes, right

My son is 12, my daughter 8. My wife and I are Star Trek Fans, so it's part of our life, and our children grow up with it.
Do you know "Trekdinners" in America? Every month, Star Trek fans meet in locations like restaurants. There are many of the in Germany. I even met my wife there.
Last weekend, we had the christmas Trekdinner with the "Klingolaus", the klingon Santa Claus. So my daughter had an little poem for him. In the summer, we have "Grilltrek", a barbecue Trekdinner ("grillen" = barbecue in german), where we all bring our children.
So, Star Trek is something, they grow up with.
Years ago, I played the Star Trek Costumizable Card Game, so I still have lots of cards. My son saw them and wanted to learn the game (good way to learn english

). Now he's more playing Magic the Gathering.
Like Little Drummer Girl already said: I'm reading the Star Trek Novels for them. 20 minutes reading for each, before they go to bed.
With Julian, I started the Destiny trilogy, then Typhon Pact, Corps of Engineers, now Double Helix. And this summer, I started reading the Academy books for Sarah (around her eighth birthday).
Both have seen a few episodes, when they were on TV, but not really much.
Early this year, Julian watched Discovery, then the 13 movies, now he started DS9, where he just finished the first season.