OK, now i would like to NEVER see anyone making such remarks about safety not mattering in a cab again.
CRTs hold a charge of up to 40,000 Volts. That's not important, it's the current. And that can hold a VERY large current, enough to kill. When using such high currents/voltages, DON'T take ANY risks. I'm not being funny. Its not just around the terminals, it can actually be anywhere. And YES it DOES REACH THE CIRCUIT BOARD!!!!! That's crap, as if the current would magically drop between the two. The circuit board is what converts mains voltage up to that required.
First, discharging the monitor. Well, Get 2 plastic or wooden-handled screwdrivers. Basically you have to hook one up to a ground point to send the charge from the HV terminals to earth. I wouldn't really recommend this if you don't know what you're doing.
Now, the other danger with CRTs is implosion. You don't really want that, imagine having your hands sucked in with a loada shattered glass.
Well, to clean you would be best off using either a special CRT cleaner or just a damp cloth. However, that "crud" is just a bit of oxidization from the HV terminals, shouldn't be too much to worry about.