What are the ranks in a us navy battleship crew?

Battleships (old and Iowa-class) are no different than other large ships like Cruisers and Carriers when it comes to ranks and rates. The only difference is that they had more crew than most.

Ship manning requirements are based on the ship's type and class, her systems, size, and tactical mission. While there's always a standard manning profile, it can change with any given mission, taking aboard more or leaving some in port as required.

For a look at the old battleship ranks/rates, you can look at the Arizona Memorial casualty roster - it shows the names, ranks and rates of all sailors who were killed.