ShepherdsDog ShepherdsDog:
Sink the bastards, rather than arrest them and you'll see piracy taper off. You can't go to a street fight and expect to win if you're the only one playing by the Marquess of Queensbury rules.
Sink them and take every ship back by force. Launch raids on the villages that are holding hostages, kill anyone holding a gun and then get the hell out (with hostages if possible).
The pirates are doing it because they can make money, if the world refused to pay ransoms and kept using force pirating would most likely end.
Another solution is convoys. Convoys aren't as fast but they would keep the ships safe.