One way is to play with the family's sense of security. Start with a normal family camping trip. Then, introduce a small, seemingly harmless event like a family photo that suddenly has a strange figure in it that wasn't there before. As the family tries to make sense of it, more and more things start to go wrong. Their food goes missing, their tent starts to leak even though there's no rain. The family realizes they are not alone and something malevolent is targeting them. This slow build - up of unease and the violation of their normal camping experience makes the story scarier.
You can make a family - centered campfire story scarier by using vivid descriptions. For example, instead of just saying 'it was a dark night', you could say 'the night was as black as coal, with only the faint glow of the campfire piercing through the inky darkness'. Also, add elements of the unknown. Maybe there's a strange sound that the family can't identify, like a low humming that seems to come from everywhere and nowhere at the same time.