My oldest brother is as big a sci fi fan as I am (he became a fan of DS9 before I did), and one of my sisters and her husband & kids are pretty big fans, too - I saw the Abrams movie with them and we all had a great time.
My dad sat me down on his lap and we watched TOS during it's original run in 1966. It's the only memory I have of my father (he would pass away in 1970 when I was 7 years old.) My mom was a huge fan. She thought Dr. McCoy had the sexiest blue eyes and a tremendous crush on Worf for his huge, broad shoulders. With mom, dad and all the older relatives gone, that leaves just us kids. Me and all but one of my cousins are huge Star Trek fans and Star Wars fans. She is in hell every Christmas because we play nothing but Star Trek episodes during our family parties and when we talk Trek she doesn't have a clue what we're saying!
Depends on the family member. My dad introduced me to Star Trek way back in the sixties, but I'm not sure my mother even knows the difference between Star Trek and Star Wars (although she proudly displays my books in the living room). As for my small army of siblings (and nieces and nephew), you've got the whole range from total indifference to casual interest to serious enthusiasm. By coincidence, my youngest brother started exposing his kids to TREK just this week. I've been helping him pick out kid-friendly eps of TOS like "Arena" and "Shore Leave." (I gather the Salt Vampire in "Mantrap" caused some bedtime issues!) As for friends . . . well, I've spent pretty much my entire adult life in scifi fandom, so the vast majority of my friends are heavy-duty fans.
My siblings were casual fans. My father was aware of the show and would tease me about it. My mother had no clue about Trek and SF in general. My wife is a fan and watches with me.
My brother called me a Star Trek nerd well into adulthood. In 2009, we saw the new movie together. It was his way of showing he cared. Now my brother is a big fan of Star Trek 2009! One day, my brother asked me which Star Trek had "the guys that wore NASA outfits"? Turns out he had caught an episode somewhere, but didn't remember the name of Enterprise. He was trying to find the series on iTunes for download. He also started calling me about a TV series he was watching. He kept asking me about Star Trek references he didn't understand. I soon started watching this show, just to anticipate his Trek questions. Now we are both big fans of The Big Bang Theory!
My sister is the only one who likes Trek in my immediate family. In fact I think she has watched all of it whereas I've only seen around 2/3rds as I've probably only watched around 2 seasons of TNG and Voyager each. My mum introduced me to Trek, but I think it was more out of that she thought I'd enjoy the TOS films rather than her actually liking Trek.