This is the easiest of all the dungeons. In fact, it's more like a stroll through the park. All the enemies are a piece of cake, and you just can't get lost.
Head up through the doorway, and left to the spring. The spring will pump you up by restoring all of your HP and MP. Next, head up to the stairs. After a few steps, you'll see a scurvy pirate hit some sort of switch on the wall. Dash over and snatch the Leather Cap out of the dungeon's only treasure chest, equip it on Reina, hit the switch, and head up through the sunlit door.
Once outside, you see a ship sailing, but how can it be sailing if there's no wind? Save your game if you want to, and then head back into the cave. Follow the path to another door to discover a large cavern. It appears to be some sort of indoor harbour for pirates! The pirates are all asleep (you can even tease them if you're really bored). There is also another switch on the far left wall. You can flip it, but the door it opens can't be reached just yet. Keep it in mind though.
After Galuf and Reina convince Bartz of their genius plan to steal a ship from under the noses of the pirates, apparently the only way left to the Wind Shrine is to head up and then cross the long bridge to the right. Once on board, go to the stern of the ship (head up), and check the steering wheel. Now you get caught, but don't worry as it's in the script.
You're introduced to yet another character named Faris, the captain of these salty dogs. Reina tries to plea with the captain by announcing her royalty (pretty stupid, Reina). Faris shackles your group anyway (with ransom demands in mind, no doubt), and throws you in the brigade. He does, however, notice Reina's pendant gleaming in some sort of light. After much contemplation by Faris, he decides that Reina's pendant and his own are very much alike and that he wants to go with your group. He introduces his pal Hydra, the water dragon, to everyone and, thusly, the windless sailing vessel mystery is resolved. Onward and upward!