Is there any use for the small crystal cave on the starting island?

On the starting island, you can go to the beach and enter a small cave full of purple crystals.

However, there seems to be nothing there besides two easy monsters.

Is there any use for the cave at all? Do I need it for a quest, or is there some item there that I haven't found?

There is something in that cave if you look around carefully. A torch might be helpful to see it.

Somewhere to the right side of the cave (from where you enter it) is a Hero's Crown, which gives you a permanent increase to some defensive stats.

You can find a treasure in there. But first you'll need to get a corresponding quest. Go to the room to the left of Carlos and open chest in it - you'll find a map.