Hopefully, my amateur explanation will not incur the wrath of anyone more professional, and please feel free to comment or correct me if I'm mistaken.

All ships, including those in present day, leak. Water pools in the bilge of the ship and has to be pumped out regularly to keep the ship afloat. In the age of sail, this was a near-daily task on most ships. So any ship of sufficient size had onboard pumps that could (generally) both pump out the bilge and pump seawater up onto the deck. In a fire, it could be fought thusly.

Having said that, fire was one of the (if not the) primary fears of ship captains. Naval ships had incredible discipline around the storage of combustibles like gunpowder and the use of flame on board the ship (slow matches, which were basically fuses used to fire the cannons, were hung over barrels of water so they wouldn't accidentally set anything alight). Shore batteries that could fire heated shot (cannonballs made red hot in a furnace, thus setting a wooden ship afire with a hit) were true menaces, as were fireships (old ship filled with combustibles and set alight, rudder and sails set to go into a crowded harbor).

TL;DR: Avoided it as much as they could, pumped water when they had to.