When discussing a question like this, we must be aware that we are generalizing and that there was no one family structure that was common to all Americans in the '60s. However, we can talk in terms of what most families were like.
The '60s were still (particularly early on) a traditional decade in terms of family. The expectation was that a family would consist of a husband and wife and their children. The father would work and the mother would be a housewife, particularly when her children were young. This was generally true in the 1960s as only 18.6% of mothers with children under 6 were working outside the home in 1960. This rose during the decade, however. It is also true that divorce rates rose during the '60s as well.
The '60s were still a time of fairly traditional family structures. However, with the coming of the women's movement and later the counterculture, this traditional structure was starting to break down.