Is it better to keep bedroom doors open or closed in the winter?

It depends where your heaters are located i suppose.
Id stay closed and that might stop the draft, but maybe consider moving your bed away from all windows if this js your concern. In the hospital i was in the rooms were warmer than the hall cause they have a heater in the room but not in the halls. If yoj have central heating with heat vents in the middle of the house it might be different but id say maybe now, it must just be draftier. My vote is doors closed unless you have reallh good central heat