Does the Catholic Church teach that the state has a positive role in society?

According to the Catechism of the Catholic Church, it does, as this selection from the Catechism makes clear:  Article 2: Participation in Social Life. Unlike the libertarian ideology (itself a substitute for religious faith), Catholicism affirms that government has a necessary role to play in protecting human dignity and defending human rights. "Every human community needs an authority in order to endure and develop," as the Catechism states.  At the same time, the Church teaches that government's power is not unlimited.  As the Catechism expressly states:

The diversity of political regimes is morally acceptable, provided they serve the legitimate good of the communities that adopt them. Regimes whose nature is contrary to the natural law, to the public order, and to the fundamental rights of persons cannot achieve the common good of the nations on which they have been imposed.

In short, Catholicism believes in the legitimacy of the state and in an active but limited role for the government in human life. Catholicism diverges from the modern liberal view of the state by insisting on limits on state authority bounded by natural law and the common good.  Catholicism also diverges from the libertarian view of the state by insisting on the legitimate authority of the state in upholding natural law and the common good.  Deo gratias.

Related item: Mark Shea reminds us that in this post that "The Gospel is Not a Political Programme." Something that bears reminding. While the Catholic Church does not propose a definitive model of government or economics, it does incorporate fundamental principles about natural law and its role in secular law and government, as my post above indicates. Within the ambit of the natural law, however, there are a wide range of possible political & economic systems. In its consistent teaching, the Catholic Church upholds both the natural law and the positive & limited role of government in effectuating the principles of natural law in the public square.