Moguls, models and members of Broadway’s elite gathered in New York City in March 2014, raising glasses in their black-tie best to toast a centuries-old student organization affiliated with Harvard University.

The club’s chair, Andrew Farkas, a billionaire and Harvard alumnus, posed with celebrities and then took the podium to welcome the evening’s guests.

Among them, in a place of honor, was a table for Jeffrey Epstein. Mr. Epstein had donated so much to the organization, the Hasty Pudding Institute, that he was listed that year as a significant sponsor and granted a table for eight guests, despite having been convicted six years earlier as a sex offender.

Mr. Epstein opted not to attend the fund-raising dinner — he notably avoided social events outside his own home. But he sent a contingent of evening-gown-clad young women in his place, according to emails released earlier this year by the Justice Department.

“sorry i wont be at your dinner, take care of my puppies,” Mr. Epstein wrote to Mr. Farkas, who responded: “I will miss you. Your puppies will have a ball. Tell them to come to the after party.”

The two men had a long, devoted friendship, which Mr. Epstein used to maintain a foothold at Harvard University years after the school banned him from making philanthropic gifts in 2008.

Now, Mr. Farkas, and the student institution he has led for over a decade, have incited anger among some at Harvard over accountability for Mr. Epstein’s crimes. The controversy taps into a broader debate about whether Mr. Epstein’s friends and associates should face consequences for having maintained ties with him after his criminal behavior was known.

The Hasty Pudding Institute of 1770 is an idiosyncratic cultural organization made up of the nation’s oldest social club, a theater ensemble and an a cappella troupe. With its long history and elite status — membership to the roughly 150-person club is invitation only — the institute has no real peers, even among the nation’s most selective schools. It has deep ties to American history, politics, Broadway and Hollywood; five U.S. presidents have been members.

It operates independently from Harvard University, is governed by a five-member board of unpaid alumni volunteers and raises its own funds.

Still, after Harvard banned his donations, Mr. Epstein saw the organization as a central way to stay connected to the school. Mr. Farkas, who has been Hasty Pudding’s chair since 2012, was his main conduit. He solicited at least $375,000 for the institute from Mr. Epstein between 2013 and 2019.

Mr. Farkas has never been accused of any criminal behavior or charged with any crime. But some students and alumni say Hasty Pudding should cut ties with him because of his friendship with Mr. Epstein.

At least 175 people — including dozens of professors, at least three current or former department heads and all of the student leaders of Hasty Pudding Theatricals from the past academic year — have signed three petitions circulating over the last several months. The petitions demand that Mr. Farkas be removed from Hasty Pudding’s leadership and ask for Harvard to strip the Farkas name from a campus building that houses Hasty Pudding and university classrooms.

Hasty Pudding’s leadership and Harvard administrators have not addressed the demands, except to point out that Hasty Pudding is technically separate from the university. James Chisholm, a spokesman for Harvard University, said that the university played “no role in its management or governance.”

The university said it had not received any official requests to remove the name from the building, referencing a formal renaming process that was recently instituted by the Faculty of Arts and Sciences.

In 2011, Mr. Farkas gave an endowment — the amount was not disclosed — for the brick Georgian revival building that houses Hasty Pudding’s theater productions as well as the school’s theater, dance and media department.

In return, Harvard renamed the structure Farkas Hall, an honor Mr. Farkas said he had chosen for his father, Robin Farkas, a businessman and Harvard alumnus who died in 2018.

Mr. Farkas is also a critical investment partner for Harvard. He has overseen investments, which have not been previously reported, that are worth tens of millions of dollars for the university, according to tax and Securities and Exchange Commission documents. The documents show several dedicated investment entities created and managed by Mr. Farkas but owned entirely by the university, including a private commercial real estate vehicle created for the school in 2014 with a $10 million seed donation from Harvard.

Tax documents show that Harvard has also invested directly with Mr. Farkas’s larger company.

Many elite universities have endowments funded by investments tied to their graduates; at Harvard, more than 70 percent of investments are with funds managed by alumni.

Representatives for Harvard, when asked about its investments with Mr. Farkas, said that the school did not publicly disclose its individual investments or managers. Through a representative, Mr. Farkas said that his company fully complied with all disclosure requirements and did not discuss client investments.

In a statement, Mr. Farkas said he was proud to support Harvard and Hasty Pudding, adding, “Naming Farkas Hall after my late father is and always will be one of my proudest moments.”

Evan Nierman, chief executive of the public relations firm Red Banyan, said that the release of the Epstein files had “created a major reputational hazard for both public figures as well as more private individuals who merely intersected with Epstein.”

His firm does not represent Mr. Farkas but has represented several people who crossed professional paths with Mr. Epstein. He argued that people should not be punished for “being convicted in the court of public opinion.”

“Guilt by association has run rampant,” he said.

But some members of the Harvard community say Mr. Farkas is different. Aidan Golub, an alumnus and former member of Hasty Pudding’s theater group, argued that Mr. Farkas’s friendship to Mr. Epstein, and the evidence in the files, should disqualify him from leading the institute.

“There needs to be acknowledgment of an awful mistake,” he said in an interview, adding, “There needs to be accountability.”

Mr. Farkas’s ties to Mr. Epstein first came to light in 2019, shortly before the sex offender’s suicide in a New York City jail, when it was reported that the duo shared ownership of a marina in St. Thomas a few miles north of Mr. Epstein’s private island.

Mr. Farkas has sought to downplay the friendship. But thousands of emails and dozens of photographs recently released by the Justice Department show they were close, sharing deep confidences, influential political contacts and crude misogynistic jokes.

Mr. Farkas was one of only a small handful of friends who traveled to Palm Beach to visit Mr. Epstein during his 13-month jail sentence for sex crimes in Florida in 2008 and 2009.

As Mr. Farkas’s connections to Mr. Epstein have become public, some members of Harvard’s community have written letters urging Harvard to reconsider its relationship with Mr. Farkas. They have resigned in protest from boards and made statements to the news media.

In February, undergraduates in the theatrical group gathered 42 signatures demanding Mr. Farkas resign as chair. In the past three weeks, two new petitions, one from affiliates of the theater, dance and media department and one from Hasty Pudding alumni, were sent to the Hasty Pudding board and the president of the university.

The pleas are not new: Multiple faculty members sent letters in 2021, when Mr. Farkas’s connections to Mr. Epstein were first emerging.

Several students and alumni interviewed for this article said the club had not responded to their concerns. One alumnus, a film producer named Michael Roiff, resigned from the Hasty Pudding theatrical’s advisory panel in frustration.

In an interview, Mr. Roiff said Mr. Farkas and the Hasty Pudding board had rebuffed his pleas for them to address the organization’s connections to Mr. Epstein. In an email, Mr. Farkas refers to the pressure as a “witch hunt.”

“The idea seemed to be that we should stay quiet because his money helps the thing exist,” Mr. Roiff said.

Hasty Pudding’s governing board did not reply to multiple emails requesting comment.

A university spokesman said Harvard was reviewing its ties to Mr. Epstein, including “some donors,” but did not elaborate.