Mr. Cuomo, a Democrat, had two emotional fulcrums in his sprawling 78-minute address: gun violence and Hurricane Sandy. But most of the speech was devoted to an onslaught of proposals favored by the left wing of his party.

He proposed increasing the minimum wage to $8.75 an hour from $7.25 an hour, public financing of elections, tougher greenhouse gas standards, solar jobs programs, a $1 billion affordable housing initiative, grants for schools that extend school days and a 10-point women’s rights program that garnered loud applause for its provisions strengthening abortion rights laws and enacting equal pay legislation.

“We are a community based on progressive principles,” the governor said, in a speech to several hundred lawmakers and guests at an auditorium in the Capitol complex. “We must remain that progressive capital of the nation.”

How the state would pay for any new programs was not clear.