First off, I've poked around in some CRTs before discharging them and didn't suffer any consequences (that I know of) whatsoever. I've wiped dust off the flyback transformers, back glass, and other random wires of a Dell P992 (which I'm still waiting on a part for btw) and on the ancient bolted-in CRT of a classic Taito cocktail (the infamous Stratovox cabinet), no big deal. Maybe I shouldn't have done this without discharging them. Doing it at some point was still a good idea, though, because that oily super-condensed dust can (so Wikipedia tells me, and it stands to reason) short out the monitor, in theory.
About discharging. I know of the screwdriver method - how does everybody else do it? A fellow named Ken Layton posted at Digital Press and I was impressed with what I read. I'll just copy that in, but first, here's a guide with pictures on discharging monitors without screwdrivers. I'm a little surprised, but it doesn't take a lot of dough to get the multimeter and high voltage probe, easily less than $60 for both parts.
Anyway, Prof. Layton's post:
Ken Layton wrote:[...] If you are going to work on the monitor main board, you should discharge the monitor's high voltage. You should remove the boards from the monitor to work on them. Don't try the "balancing act" of trying to work on the boards while still in the monitor. [...]
If you are removing a complete monitor from the cabinet there's no need to discharge it. However, you do need to disconnect the power cable, video/sync input cable, and frame ground wire from the monitor before lifting it out. Always have a helper handy to watch to be sure all cables are clear (not snagging on anything) and to help lift the monitor. Remember that 25" monitors are heavy (heavier than they look) and you don't want to drop it or break the neck of it. If you should decide to lay the monitor "face down", put a blanket or pillow down first. The tube will scratch itself under it's own weight and that is a permanent condition.
Some people remove the monitor to do cabinet work or to install a new or better monitor.
When I discharge a monitor I use a high voltage probe to do it. This is the best and safest way. You bleed the charge off quickly in about 15 seconds and with a proper load. You can see the voltage on the meter as it's discharging and without that dangerous SNAP of doing it with a screwdriver. [Emphasis added]
Sure you can discharge the tube with a screwdriver and a piece of wire. However, that can damage things. That sudden snap discharge can arc/burn the prong contacts inside the rubber suction cup thus making a poor connection later. Inside the flyback transformer is a high voltage diode that can be damaged by this sudden short circuit to ground.
I generally do not discharge a monitor just to move/remove it from a cabinet unless it's one of those oddball "two-piece" jobs where the tube is mounted to a piece of wood and the circuit boards are screwed to the side of the game cabinet.
I've been in this business since 1976 and have literally repaired over 1,000 monitors over the years.