The decision drew protest at the board’s morning meeting.
Opponents say shutting down the hospital will leave some 2,000 people out of work.
However, SUNY Downstate – which runs the facility – says the hospital has a $40 million budget shortfall and would have run out of money by next month.
A recent report from the state comptroller found the hospital has lost money for 17 straight years.
The hospital now must submit a closure plan to the state Health Department.