The governor has spent a lot of political capital on the issue, and even brought in some star power to help. Earlier this month, Lady Gaga and Mr. Cuomo were co-authors of an editorial in Billboard magazine, and the governor has been promoting a documentary called “Hunting Ground,” about sexual assault on college campuses. The film was co-produced by his sister, Maria Cuomo Cole, and Lady Gaga contributed to the soundtrack.

Mr. Cuomo, in a statement, called the legislation “a major step forward to protect students from an issue that has been plaguing schools nationwide for far too long.”

The plan builds on a standard set by California, and extends policies in place at New York’s public colleges to private institutions in the state. It arrives as campuses across the country have come under increased scrutiny for their handling of sexual assaults. To that end, Mr. Cuomo’s plan calls for establishing a new unit in the State Police to help work with college campuses, since they are often the first point of contact when a rape occurs.

The governor’s bill also requires all campuses to distribute a so-called bill of rights to students. It will inform students that they can report sexual assaults either to campus police or to outside law enforcement, and that in that context they will be granted immunity for certain campus rule violations, including drug use.

Ms. Glick said the legislation had evolved to address concerns that the initial proposal used terms like “victim” and “the accused,” which many people felt conveyed bias. The current bill uses more neutral language, like “reporting individual” and “respondent.”