How to avoid water build-up in bathroom's passive ventilation shaft?
The problem can be solved by draining the vent. Be sure to use a p trap to prevent sewer gas from entering. Yes, the problem could be condensed moisture, but it might also be rainwater that is collecting. Either way, a chimeny cap that is shaped like a cone will cause an aspiration effect (like the negative pressure under an airplane wing) which will naturally draw air upward through the vents; plus it will make sure that rain water doesn't collect in the vent.

I would also consider adding a ball valve after the p trap, because with these corrections, I wouldn't expect to find enough water (through condensation) to fill a p trap; and sewer gas could enter through the p trap if there isn't any water in it... so have the ball valve ready (to close the drain) just in case the chimney cap solved the problem.