The idea is presumably that they 'prevent', rather than 'remove', condensation - by removing air containing a lot of water vapour before that water vapour condenses on cool surfaces.If fans are so brilliant at removing 'condensation' from a room then why do they not remove it from a duct?
Once condensation has happened (i.e. water vapour has turned into liquid water on walls etc.), there's not all that much that a fan can do to remove it, other than (perhaps) replacing air in the room with air of lower humidity, thereby encouraging evaporation - but such evaporation is a pretty/very slow process.
Yes, maybe - and, if so, that's a more fundamental problem than any 'condensation'.From the description in the OP it appears that the top of the duct is not preventing entry of water and dirt from outside.
Kind Regards, John