A simple way to avoid drowning and dying is to bring along with you several doors and blocks of glowstone or jack-o-lanterns. Glowstone can be used as an underwater light source, as can jack-o-lanterns, and doors can be used as an easy-to-use, readily available underwater air pocket, allowing for quicker mining and less deaths.
To succeed at this quest, there are two checks that must be met. First, there must be a block of glass above your head. If this check is not met (checked at 25-second intervals by the game, the results of which are outputted to the chat in the bottom-left corner) then the game does not examine the second check. If there is glass above your head, it moves onto the second check: how many blocks of water there are above your head. If the value is 21 or higher, then the quest is complete and can be turned into the Sadhu. If the value is less than 21, then it says that there are not enough water blocks and you must try again.
Once you return to the Sadhu, he rewards you with 128 reputation, 4 experience points, and the ability to move onto the next quest in the series, Limits of the World.