“There really aren’t that many people who are giving to art and giving to museums, in fact it’s a very small club,” said Tom Eccles, the executive director of the Center for Curatorial Studies at Bard College. “So we have to be a little careful what we wish for here.”

There is also the difficult question of where to draw a line. What sort of behavior is inexcusable?

“We are not a partisan organization, we are not a political organization, so we don’t have a litmus test for whom we take gifts from based on policies or politics,” said Mr. Weiss of the Met. “If there are people who want to support us, for the most part we are delighted.”

“We would only not accept gifts from people if it in some way challenges or is counter to the core mission of the institution, in exceptional cases,” he added. “The OxyContin crisis in this country is a legitimate and full-blown crisis.”

Three brothers, Arthur, Mortimer and Raymond Sackler, bought a small company called Purdue Frederick in 1952 and transformed it into the pharmaceutical giant it is today. In 1996, Purdue Pharma put the opioid painkiller OxyContin on the market, fundamentally altering the company’s fortunes.

The family’s role in the marketing of OxyContin, and in the opioid crisis, has come under increased scrutiny in recent years. Documents submitted this year as part of litigation by the attorney general of Massachusetts allege that members of the Sackler family directed the company’s efforts to mislead the public about the dangers of the highly addictive drug. The company has denied the allegations and said it “neither created nor caused the opioid epidemic.”

Arthur Sackler died before OxyContin’s creation and his side of the family, which has supported institutions including the Smithsonian and the Brooklyn Museum, sold his stake in the pharmaceutical business after his death. One of his children, Elizabeth A. Sackler, has called the company’s role in the opioid epidemic “morally abhorrent.”