I think Failmond's first post does a solid job with the analysis.
You have a voltage potential across a resistor(the air). You can not remove air from the equation as it is what is sourcing and sinking the voltage potential in question. The maximum energy you can harness is what it is already sinking.
The amount of power harnessable is indeed very low, but if I were to build a theoretical device to harness this power it would involve high altitude balloon system with lots of surface area. High altitude would mean high potential and higher the surface area the higher the current. It would be tethered with a light weight insulated conductor. The conductor could be very small gauge wire because of the very very small current, but the insulation would be very thick and have a high dielectric strength. Once the power made it to the ground it would connect to one leg of a specialized high voltage dcdc converter involving a spark gap and a transformer to convert the power down low enough to be useful. The other leg would be connected to an array of small copper wires going across the ground in all directions.
So, its possible, but at the end of the day you would produce a continuous 100 watts of power and have incurred millions/billions of dollars in expense. You have also created a really good lightning rod which would destroy the system unless even more expensive measures were taken. On a similar note, you can not harness power from lightning because it will destroy any reasonably sized system. And an unreasonably sized system, sized to handle 99% of lightning strikes without failure, would be many many many times over non-cost effective.
Its not reasonable to harness this knowledge for power production. However, in the future it could possibly be used for sensors much like an electroreceptive fish.