A ThermalSaltSedimentor is a nice system, but requires a water column covering a large range of height which might be either ugly if built up the side of a mountain, or expensive if entirely built.
Instead of having a tower, we might work this whole system in batch. In this system, a brackish salt solution is held in an AirShip, bouyant with Hydrogen. The AirShip rises to, say 5km in height, where it freezes, forcing the salt out of solution. Salt is collected at the bottom, ice at the top from where they are gathered and stored.
Now assuming that the AirShip has reached its maximum bouyant height, it can be made to sink fairly easily. Either some of the Hydrogen could be released (costing Energy), or it could be compressed (again, costing Energy)), or, perhaps, consumed in a small BioFuelGenerator. Either way, the system will start to sink.
On the way down, incidentally, the frozen water and cold salt, could actually be used to condense water out of the atmosphere. This is particularly good because a) it will be fresh and b) it will add to the weight of the whole AirShip ensuring that it keeps sinking without using up much of it's Hydrogen. Finally, it should mean that the system is practical even in areas where water is hard to come by, replacing natural wastage.
Eventually, the AirShip sinks down to ground level. The separated salt and fresh water is offloaded and more brackish water introduced.
There are a number of different arrangements that you could use to exploit this system. With smaller AirShips, they might be tethered to the ground so that they stay in one place. For a much larger system, the wind might make a tether impractical. However, much like a HydrogenBalloonNetwork, if there was enough infrastructure, the AirShip could go up one place and land in another.