Why does Utopia make the idea of family something unacceptable and unsavory?

Mustapha Mond, controller of the World State, explains more than once what is wrong with traditional families, a form of social organization he seems to particularly abhor. He states,

What suffocating intimacies, what dangerous, insane, obscene relationships between the members of the family group!

Mond also notes the following:

Our Freud had been the first to reveal the appalling dangers of family life. The world was full of fathers—was therefore full of misery; full of mothers—therefore of every kind of perversion from sadism to chastity; full of brothers, sisters, uncles, aunts—full of madness and suicide.

As the above quotes show, the World State believes family life breeds dysfunction and misery because family members are too closely involved with each other. The state has decided people are more emotionally healthy if they are raised in a communal, dormitory-type setting rather than in a nuclear family. This way, people can be conditioned to accept and appreciate their lot in life and grow up as well-socialized members of the larger community.
To prevent the formation of nuclear families, eggs are harvested from women and artificially inseminated with sperm, then divided into as many identical twins as possible—the lower the caste is, the more embryos are produced. Bearing a child is considered disgusting and obscene.
In this society, the emphasis is on social harmony. This is best achieved through fleeting, superficial relationships between people, rather than strong attachments. It is easier for the society to enact its motto that everyone belongs to everyone else if all people are raised as part of a mass, not in family units.
Huxley, of course, satirizes this form of mass social organization.

In Brave New World, the idea of family is unacceptable and unsavory because it goes against this society's key philosophies. For a start, a family (the traditional unit for the procreation of children) goes against this society's program of artificial reproduction. A family would negate the need for the Hatcheries and the Conditioning Centers where children are molded into particular types of people.
Secondly, a family is also seen as unsavory because it contravenes the sexual liberation of this society. As we see through characters like Lenina, this society also values sexual promiscuity. Moreover, women have ready access to drugs that can prevent cases of pregnancy.
Because the society of Brave New World believes that it has created a utopia, the idea of reinstating the family unit would, in theory, destroy this creation. By allowing people to form loyalties and attachments to one another and to allow couples to procreate, this society as its inhabitants know it would be completely destroyed.