Do you pull out the setting screw (crown) to stop battery drain??

Pulling the crown out so the movement doesn't tick doesn't necessarily save the battery, it depends on the movement. If you aren't going to wear the watch for some time, you are better off removing the battery

I don't like the idea of leaving the crown out in case it might be more vulnerable to moisture, dirt, etc. getting inside. I'd be more likely to remove the battery for long term storage.

I'd agree that pulling the crown may slow instead of stop power loss, but I'm very curious about this notion that disconnecting the literal motion of the watch has no effect whatsoever.

I absolutely do this on a few quartz watches that don't see enough use to justify their battery drain.

I'm not convinced that works.

I asked this question regarding my Seiko 7548 Quartz dive watch of Spencer Klein, the vintage Seiko watch specialist. He advised that pulling the crown out to the second position would stop the train of gears, as it disengaged them from the stepper motor. However electricity from the battery would still get through to the motor, draining the battery. Worse if the crown is out for a long time & the battery goes flat the low battery indicator of the seconds hands moving every two seconds won't work to warn of this. This may not be the same for other quartz movements however.

The main reason i have quartz watches are to pickup & go , pulling the crown out would defeat the purpose

I wear my quartz watches regularly and replace their batteries while they are still within their life, so battery loss isn't a worry for me. If I had a quartz watch that I wore very rarely, I'd rather just remove the battery entirely, stopping leakages (even though those take years, often decades) and power drain altogether.

Batteries cost so little though.

Seems like penny pinching in the extreme to me. Not worth the effort.

Plus, there have been stories of people doing this and humidity entered the watch & corroded the movement beyond repair.

If you store your watches at a normal dry safe place, should that not be a problem... And it's not the money... Everytime you open a watch, may you damage the rubber seal and let some dust come inside - so I think it's better to avoid open it extra times.