The correct answer is A) lowered down into the water. When rigging a stage over the vessel's side, the remainder of the lowering line should be lowered down into the water to prevent it from getting tangled or caught on anything. This is a common safety practice in marine operations to ensure the line is clear and readily available for future use or retrieval. The other options are incorrect because: B) coiling the line on the stage could lead to tangles or obstruction; C) coiling the line with the bitter end on the bottom would make it difficult to access; and D) coiling the line on deck could create a tripping hazard and would not allow the line to be readily available when needed.