The neighbors behind me recently dug a driveway three feet below my property, right up to my fence. The fence is teetering on a ledge over their driveway. If the dirt under the fence erodes and the fence falls over, is my neighbor responsible for fixing the fence? Also, if a child climbs the fence and gets hurt, who is liable?
Your first order of business is to find out whether the fence is in fact on the boundary line. If so, then you and the neighbors co-own the fence and must share upkeep costs and liability for any personal injuries the fence causes. See Boundaries and Neighbors FAQ for more information on determining where property lines lie.
At any rate, if the neighbor's action causes the fence to fall down, the neighbor should be responsible. If you are co-owners, however, then you may have to share the cost of repair or replacement.
Before getting all lawyerly on your neighbor, try this tried and true approach: Stop by your neighbor's house -- or better yet, the neighbor's yard when he or she is already outside and can look at the fence with you -- and say: "What do you think we should do about this fence? I'm worried about it."