The correct answer is D) Check bearing lube oil temperatures. The main propulsion reduction gears are critical components of a vessel's propulsion system, and monitoring the bearing lube oil temperatures is an important operational check that should be done continuously. Elevated bearing lube oil temperatures can indicate issues with the reduction gears, such as improper lubrication or misalignment, which could lead to further damage if not addressed promptly. This continuous monitoring helps ensure the safe and reliable operation of the vessel's main propulsion system. The other options, while important, are typically not required to be checked continuously. Checking radial bearing wear, inspecting alignment, and checking for pitting and scuffing on the gear teeth are important, but they are usually done as part of periodic maintenance and inspections, rather than requiring continuous monitoring.