The first principle of the State is to promote the common good. Since all members of society pass through the institution of the family, the State has a duty to promote the health of families and marriage.
Sure you don't want to rethink that? You may want to brush up on some history my FRiend...the role of the "state" in America (as it is plainly written in the Constitution) is much, much less intrusive and have less "duties" than you think.
As for the common good...see tagline.
PreambleThe preamble sets forth the purpose of the Constitution.We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquillity, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America.
As for Hilary Clinton, she's simply abusing the term.
The exhaustive treatment of the proper responsibilities of State and Church here is the best and most concise that I've seen on-line.