Councilman Ben Kallos, a Manhattan Democrat, and others have said the board should be nonpartisan. Political parties hold too much sway in the appointments, he said.

“At the end of the day when you have an institution run by patronage, where people are there because of who they know and not what they know, I will never be confident that they will be able to run a smooth election,” said Mr. Kallos, who leads a council committee that has oversight of the board.