My take was that the crucial 'energy' issue was the huge latent heat of evaporation of water that cooled the air at the top of the tower. So, the air falls in the tower partly, maybe mostly, because it is cooler and, thus, more dense, not because the air is more dense because it is wetter (is air less dense than water vapor?) but because it is more dense because it is cooler.

Then I would wonder how to recapture the water without losing a lot of energy, that is, would be fighting the huge latent heat of evaporation again? And I would wonder just why bothering to desalinate the water?