This is kind of a hard question to answer, simply because University City is traditionally thought of as being a place for college students. However, Powelton Village is technically in University City and could be a great place for a growing family. Over towards Spruce Hill is also a nice area.
I think the biggest concern for families would be the overall noise and the traffic. The influx of college students causing traffic nightmares, and the area can get really chaotic on weekends. Stay away from dorms and known rental units and you should be fine.
