The correct answer is D) motion begins to be excessive. When a semisubmersible rig is under tow, the motion of the rig should be monitored closely. If the motion becomes excessive, meaning the rig is rolling, pitching, or heaving excessively, the rig should be ballasted down to increase its stability and reduce the motions. This helps prevent potential damage to the rig or towing equipment and ensures the safety of the operation. The other answer choices are incorrect because they do not directly relate to the need to ballast down the rig due to excessive motion. The period of roll, head winds, or water breaking over the pontoons may be indicators of issues, but they do not necessarily require ballasting down the rig, whereas excessive motion does.