Well you can talk with the people of Besaid if you want, but still they will only give you clues about the things you should do. Now, to make sense of the clues:

- There is a door located where the path forks, near the beach.

- Four ciphers, a four-digit code, is needed to unlock the door.

- These four numbers are hidden around the island.

- The easiest digit to see, on the stone statue overlooking Besaid, is the first in the code.

- Two digits hidden around the shore make up digits two and three.

- The last digit is on top of shattered ruins, over the path near the waterfall.

I would advise to write down the codes and then examine the door, if you've gotten all the clues and the numbers, the correct code will appear above. Enter it. Access granted! Enter the cave.

I hope that works