There are two ferries that serve the Fulton landing at the north end of Brooklyn Bridge Park: the East River Ferry, which doesn't serve the west side of Manhattan, and New York Water Taxi, which does.
There's a ferry landing at the south end of the park, but as far as I know it's used only for seasonal service to Governor's Island.
The water taxi seems to require you to buy an all-day pass for $31, and it takes about 50 minutes to get from West 39th street to Brooklyn Bridge Park.
From Times Square, I would probably take the A train to High Street, but if I were determined to take the ferry, I would take the M34 bus and get the East River Ferry at East 34th St. The fare is much lower ($4 or $6 each way, the higher price on weekends, and an all-day pass $12 or $18); the duration is about 25 minutes, and the service is more frequent.