Ms. Arsht, a Democrat, was emphatic that where members stand politically “absolutely does not come into play” on the Lincoln Center board. She said she and David H. Koch , the conservative billionaire, had served together on the board of the American Ballet Theater in the 1980s.

“David was passionate about it,” she said. “I don’t believe there was any political baggage back then.”

Conservatives like Mr. Ross serve on the Lincoln Center board, but so do prominent liberals like the music mogul David Geffen , whose name is on the building where the New York Philharmonic performs. (Across the center’s plaza is the home of the New York City Ballet, a building named for Mr. Koch, who is a director emeritus of Lincoln Center.)

A spokeswoman for Lincoln Center declined to comment.

Ms. Arsht noted that the board of the Kennedy Center for the Performing Arts in Washington was politically split by design: Its members are appointed by the president of the United States. Mr. Trump has appointed spouses of several real estate developers he knows, as well as Mike Huckabee, the former Arkansas governor and father of Mr. Trump’s former spokeswoman, Sarah Huckabee Sanders.

(Ms. Arsht was appointed by President Barack Obama, but Mr. Trump renewed her term.)

The world of philanthropy has seen a few protests. After Mr. Koch paid $65 million to renovate the entrance to the Metropolitan Museum of Art in 2014, it was named for him. The reopening drew protesters, who were largely focused on Mr. Koch’s stance against climate change.