Several universities on the East Coast announced new policies over the weekend and Monday to head off the spread of the coronavirus, including virtual classes and curtailing campus gatherings.

Their measures came after major schools took steps to curb regional outbreaks on the West Coast. Last week, the University of Washington shifted entirely to online courses for its 50,000 students, and Stanford University announced that classes would not meet in person.

In New York, four universities followed suit over the weekend. Columbia University canceled classes Monday and Tuesday and said that classes would be taught remotely for the rest of the week. Hofstra University on Long Island and Yeshiva University in Manhattan announced that classes would be canceled for the week.

On Monday, two more schools announced similar measures. Princeton University said in a statement that it was requiring all lectures and seminars be held virtually, starting March 23, and that it would curtail large gatherings on campus. And Berklee College of Music in Boston announced that students traveling over spring break to certain states and countries would not be allowed on campus until after a 14-day self-quarantine.

A student and two faculty members at Yeshiva tested positive for the virus last week, and Hofstra said that a student had started showing symptoms after attending a conference; that test result is pending. Columbia said a university community member was quarantined for exposure to the new coronavirus.

Princeton also said it would restrict “university-sponsored travel,” and encouraged students to stay home after spring break, which runs through March 22. The university said that the risk to the school was low, but that the restrictions, which will be in place until at least April 5, were proactive.

“Our medical advisers tell us the best time to put in place policies to contain the spread of #Covid-19 is now, before we begin to see cases on our campus,” the school said in a tweet, referring to the virus.

Berklee said on its website that anyone returning from countries where the outbreak is particularly sever — South Korea, Iran, Italy and China — or from states that have declared emergencies — California, Florida, Kentucky, Maryland, New York, Oregon, Utah, and Washington — would have to self-quarantine for two weeks.

In a post on the university’s website, Columbia’s president, Lee C. Bollinger, said it was important to note that the person who had been exposed had not tested positive for the virus.

“This action is intended to prevent the virus from spreading,” Mr. Bollinger said.

Mr. Bollinger did not say whether the person was a student or a faculty member, or how the person had been exposed to the virus.

Gov. Andrew M. Cuomo of New York declared a state of emergency on Saturday as the number of coronavirus cases rose. As of Sunday night, more than 100 people in New York State had tested positive for the virus, according to state and local health officials, as well as the Centers for Disease Control and Prevention.

Barnard College, which is considered one of Columbia’s four undergraduate colleges but is an independent institution, announced that it was taking the same precautions. In a post on Barnard’s website, the college’s president, Sian Leah Beilock, characterized the person’s exposure as an isolated case.

The cancellations coincide with midterm examinations for many students, which Ms. Beilock said she had asked to be suspended or given in a take-home format.

“It is important to know that our actions are out of an abundance of caution,” she said.

Students at Columbia and Barnard College won’t return to the classroom for two weeks because spring recess is next week. In 2019, there were 33,413 total students and 4,364 total faculty members at Columbia, according to the university’s website. At Barnard, there were 2,682 undergraduate students and 253 full-time faculty members as of the fall.

Kelly Fuller, a freshman at Barnard who is currently taking a global health class, said it was difficult to say whether canceling classes was the best decision, but “I’m glad they’re doing something.”

“When we got the email, it was like, thank you, yeah, that’s what I wanted,” she said. “You’re recognizing that it’s important for us to not get sick.”

Maya Perry, 22, called it a “sensible” decision, but said that all of her classes at Columbia this semester were small-group seminars, like a 12-person class on American cultural criticism that she worried might not work as well online.

“I was looking forward to a particular class discussion on Tuesday, but obviously safety is more important than that,” she said.

Ms. Beilock said that residence halls, libraries and other university buildings would remain open, but that all events and gatherings had been canceled, in addition to admissions tours. She said the university had consulted with city and state health officials on the plan.

“We all share the goal of upholding the health and safety of our community,” she said, “and we ask for your consideration and caring of each other during this difficult time.”

The precautionary measures taken by Columbia and Barnard came as public school administrators in one town in Westchester County, N.Y., announced on Sunday that a middle school teacher had tested positive for the virus and that classes had been canceled for the week at all public schools there.

The town, Scarsdale, which is about 25 miles north of Midtown Manhattan, was following the recommendation of the Westchester County health commissioner, school officials said in a post on the district’s website.

They said the teacher was showing mild symptoms of the illness and that the school district wanted to be vigilant for “immuno-compromised” people.

The school system said it was working with health officials to contact families of students who had been exposed to the teacher. All school buildings will be disinfected, the post on the district’s website said.

“We recognize that this news will be concerning to many of our students and families,” the post said.

In New Canaan, Conn., public school officials announced on Sunday that classes would be delayed two hours on Monday so teachers could organize materials for students to take home with them in case the schools close because of the coronavirus.

Ned Lamont, Connecticut’s governor, announced on Sunday that a resident of Wilton, Conn., which neighbors New Canaan, had become the first state resident to test positive for the virus. The person, who the governor said was between 40 and 50 years old and had recently traveled to California, was being treated at Danbury Hospital.

A New York resident who is an employee of Danbury Hospital and Norwalk Hospital in Connecticut also tested positive for the coronavirus, Mr. Lamont said on Friday.

Daniel Victor contributed reporting.