When may passengers legally ride on the bow or gunwale when the vessel is underway?

If your boat is a bow rider, that is it has seats built in to the bow are, and hand holds or railings, and people can sit safely, without risk of falling over board, or blocking the operators view, then they can ride there. When a boat is coming into the dock and someone needs to go forward to handle lines, but only if the boat is moving slowly and it is safe to do so. On sailboats to handle or change sails. On either to raise an anchor.

On the gunwale, only when the boat is not moving, or it is necessary to retrieve lines or an anchor. The exception is some inflatable boats where the only place to sit is on the tubes, and they have handles or hand holds to hang on to.