My comments probably echo most of the other responses.
In training (PADI), I was taught to turn the air off and to purge the system after equipment assembly if I'm not getting directly in the water.
A reasonable approach. I teach that to OW students as well. It prevents air loss either through a very slow, small leak (e.g. a yoke cylinder O-ring) or having a significant purge occur because something (e.g. someone else's gear) inadvertently cause a purge valve on one of my second stages to be depressed (but not audibly apparent, so it goes un-noticed).
t-mac:
In all honesty, sometimes I do and sometimes I don't. I think the reason for my inconsistency is that I'm not sure I have been given a compelling reason to turn the air off.
The inconsistency is a bigger issue than whether you leave the valve open, or close it. There is no reason to be inconsistent. Either you do it (close the valve) because you were trained to do it, or you don't do it because you don't believe that there is a compelling reason to do it. Frankly, either practice is OK, but the inconsistency is simply illogical.
t-mac:
In fact, it seems to me you increase the risk of getting in the water with your tank turned off if you do this (especially if you don't purge it).
That argument could be made, I guess. But, getting in the water with your cylinder valve closed really has nothing to do with the issue of closing the valve, or leaving it open, after assembly. Rather, it has to do with a poor / incomplete pre-dive check. I check my gear immediately before entering the water, whether I have assembled it 10 minutes before, or two hours before. The assembly procedure and the pre-dive (safety) procedure are two entirely different activities, and should be treated as such.
t-mac:
I have been told that I could damage my regulator if it is left pressurized in the sun -- compelling, if true!
And, there are no objective data (and therefore no compelling reason) to support such a belief.
I have also heard instructors use the argument that it is better to find out that you have a slow leak before entering the water, therefore it is better to leave the valve open. I understand the logic but don't subscribe to it. I would rather start with a full cylinder, monitor my pressure after splashing, and then end the dive early if I find my pressure is dropping more rapidly than expected, than a) either start with a half-empty cylinder, or b) not splash at all.
t-mac:
I assemble the scuba unit, pressurize the system, check my cylinder pressure, check the function of my second stages and the inflator, then close the cylinder valve and purge the system. When I get ready to enter the water, AND I have donned my scuba unit (after opening the cylinder valve), I go through a pre-dive safety check in which I (again) check the cylinder pressure, and function of the inflator and both second stages. For me, doing it this way consistently for each dive is the best approach.