Like anything else, it can be either a ma’aleh or a chisaron. If a person takes the lessons of his/her family to heart, and strives to live up to the family reputation, then it’s a ma’aleh. If (s)he uses it as an excuse to do and get whatever (s)he wants, it’s probably not such a good thing.
How important is it? As important as you make it. If you are specifically looking for yichus, then yeah, it’s important and should take precedence over something you don’t care about as much. It all depends on what you’re looking for.