Would you save energy by heating the air in a shower stall so that you could use colder water? It is refreshing to take a cool shower in hot weather. And for the sake of discussion, lets assume that one should be "comfortable" with temeratures when taking a shower.
Considering that the vast majority of the heat from a shower is lost down the drain, would it not be more energy efficient to heat a confined space (shower stall) to a temperature that would allow you to shower with cool water rather than hot?
How hot would it have to be in the room for 70 degree water to be "nice." 60 degree?
And since all of the energy used to heat the air in the space would remain in the space/room, would you use less energy to take a comfortable shower doing it that way?
 A: Consider that the specific heat of water is 4.147 kJ/kg. The specific heat of air is less than 1/4 of that, so for the air to heat you as much as the water, you would have to heat the air to a much higher temperature. That's one reason, along with conduction, that Arctic explorers say that water cools you 30 times faster than air.
So it doesn't balance out. It's 4 times easier to heat air, but water cools you 30 times faster. 
Even if this wasn't the case, your system would need to have a much higher temperature in it, which would mean more electricity, which would mean more electricity wasted in wires. 
A: According to a steam shower vendor, a 10KW unit is required to provide enough steam at 118 deg.F in a 6'x8'x 8' enclosure. The timer runs for 20 minutes (about right for a nice shower), so figure the unit would use 3.3KW and 2 gallons of water to make the steam room hot and steamy.
According to numbers pulled from all over, figure an average 20 min shower uses 50 gallons of water (416.5 lbs.), and that in my area that water comes out of the ground at 55 degrees F. To heat that water to 110 deg.F would take approx 22,910 BTU.
Converting KW to BTU (and here are the limits of my understanding - please correct my conversions/assumptions as needed) it looks like 3.3KW is about 11,260 BTU.
If this is all reasonable, then heating the shower to 82.5 deg.F would require 11,500 BTU, which, when added to the 11,260 BTU for the steam would total 22,760 BTU - or slightly less than just running the hot shower by itself. If you could go with heating the shower water even less,then the savings would appear to increase.
I am fully aware that this answer is full of broad assumptions, simplifications, and perhaps misunderstandings.
