“Unethical and even criminal wrongdoing has badly damaged the reputation of this agency,” Mr. O’Toole said, in explaining the censure. “We will not go back to the way it was.”

The censure was unanimously approved, across state and party lines. Ms. Turner had abruptly resigned from her Port Authority seat last week after the video surfaced, making the public censure largely symbolic.

Ms. Turner, like Mr. O’Toole, was appointed to the board by the previous governor of New Jersey, Chris Christie, a Republican. The board also approved a referral of the matter to the New Jersey State Ethics Commission.

Daniel J. Horwitz, a commissioner appointed by Gov. Andrew M. Cuomo of New York, said the referral was necessary because Ms. Turner, in a statement about the episode she released publicly, denied that she had violated the Port Authority’s code of ethics or had asked for special treatment.

“Anyone who’s seen that video will agree that the conduct on that video is not only out of bounds for any public official, it’s out of bounds for any member of the public in respect to how they act with the police,’’ Mr. Horwitz said.