I'm sure that everyone in EVE has heard by now of the wonderful bait-and-switch move that GoonSwarm pulled on Band of Brothers a few months back. (For those who haven't, the summary is that Goon infiltrated BoB at the Director level of their receiver corporation and then proceeded to dismantle BoB from within) Corporate CEOs and Alliance leaders both look at this with no small degree of fear, because it illustrates greatly what happens when you lack good corporate/alliance security.
The concept of having your corporation or alliance destroyed from within by a high-level betrayal is the one concept that keeps CEOs awake at night. You never want to wake up and log on to EVE only to discover that the corp coffers have been cleaned and the corp fleet of shiny battleships or capital ships are now missing. But, as CCP has shown in the past, corporate espionage is as much a part of the game as anything else.
There are several ways for CEOs to help ease their nightmares and sleep more peacefully (without having to resort to alcohol or sleeping aids... or both).
First and foremost, know your Directors and trust them. No one should have the Director title within the corporation unless you know them well enough and have vetted them enough to trust them with your life (literally in game) and livelihood.
Second, have good practices in place for handling applications from prospective members. There are many ways to go about this - from requesting API keys for verification to screenshots of the EVE login screen to catch alts. As well, having probationary periods in place is a good idea - it shows the applicant that your corporation is both serious about them and takes their security seriously.
Third, pick someone trusted to handle your Internal Security work. This person should be suspicious by nature. Give them the appropriate roles and have them perform audits periodically to make sure that nothing is going on that shouldn't - people aren't receiving any roles or titles they shouldn't and things aren't appearing mysteriously in member hangars.
Fourth, handle corp goods with care. There should be one corporate hangar that only Directors and above have access to - and even there all items should be in secured containers. This will limit the amount of petty larceny that goes on within the corp hangars.
These are just basic ideas but implimenting them in one form or another can save a CEO lots of headaches down the road and reduce the chances of falling victim to corporate espionage.