One new idea for revenue is a 2 cents-per-milligram surcharge for the active opioid ingredient in prescription drugs, a root cause of the heroin epidemic that has seized the state and nation. The governor is also proposing an “internet fairness conformity tax,” a holdover idea that would require so-called “marketplace providers” like Amazon.com to collect sales tax. (It failed to win approval last year.) He reiterated a call for the closure of the carried interest loophole, a federal tax that allows some hedge fund managers to pay a lower tax rate on revenue from investments.

John J. Flanagan, the Republican leader of the Senate, praised elements of Mr. Cuomo’s plan, including a 3 percent increase in education funding, the largest chunk of state spending. He did not seem enthusiastic about the payroll tax idea, calling it “very challenging,” though he said he had several discussions with the governor about the issue.

In a small surprise, Mr. Cuomo also said that he would ask the Department of Health to study how New York would be affected by neighboring states like New Jersey and Massachusetts legalizing or moving toward the legalization of recreational marijuana. The stance seemed to be a shift for Mr. Cuomo, 60, who was slow to embrace medical marijuana in the state, though he signed a limited legalization of such use in 2014.

Having been stung by bad news in the city’s subways during 2017’s so-called summer of hell, a phrase coined by the governor, Mr. Cuomo is proposing an “action plan” of $254 million for the Metropolitan Transportation Authority. This would be money taken from financial settlements and the accelerated transfer of funds from a tiny payroll tax paid in the 12-county M.T.A. service area.

The proposed opioid tax would raise an estimated $127 million in the coming fiscal year, while another plan — adding a 14 percent surcharge on profits made by health insurance policies — could raise $140 million, according to administration estimates.

The state’s budget bills that are slated to be approved by April 1, the start of the fiscal year, are often a vehicle for nonfiscal policy, and this year was no exception, with the actual budget bills packed with issues like same-day voter registration and bail reform. Mr. Cuomo used the speech as a platform to propose a range of changes to sexual harassment policies in the state in light of the #MeToo moment; he called it “a national crisis,” which was given new life in Albany last week when a powerful senator, Jeffrey D. Klein, was accused of forcibly kissing a staffer in 2015. (Mr. Klein was in the audience, and has denied the allegation.)

The addition of sexual harassment to the budget process may create an awkward optic: There are no women among the four men who are charged with negotiating the budget — Mr. Cuomo, Mr. Flanagan, Mr. Klein, and Carl E. Heastie, the speaker of the Assembly. For his part, however, Mr. Klein said on Tuesday that he expected to be vindicated. “We have to strengthen our sexual harassment laws,” he said. “And I intend to push to make that happen.”