The correct answer is C) Master of the vessel. The master (captain) of the vessel is responsible for ensuring that someone is assigned to close the watertight doors in an emergency. This is because the master has the overall command and control of the vessel and is accountable for the safety of the crew, passengers, and the vessel itself. The master is required to have emergency procedures in place, including the assignment of crew responsibilities for closing watertight doors, as part of the vessel's safety management system. The other options are incorrect because the Coast Guard (A) sets the regulations but does not directly assign crew responsibilities on a specific vessel, and the chief engineer (B) and chief mate (D) are crew members who may have assigned duties but are not ultimately responsible for the overall emergency response plan.