1) You have at least one actual or potential spy in your corporation right now unless you're a one-player corp. Actual - someone who either joined your corporation to spy on you or steal from you, or who has decided since joining to do so. Potential - someone who is vulnerable to becoming a spy or thief due to frustration with the corp, grievances real or imagined, or coercion or seduction by an outside individual or group. Minimize the damage that can be done to your corp by compartmentalizing access to assets and information. At the same time, be open with your pilots about what your security procedures are and why you have them. Make them uniform, so all your pilots know that it's not because you don't trust them personally, it's a universal corporate policy that applies to everybody.
2) If you suspect a threat within your corp, don't take half-measures. An infiltrator or turncoat will be prompted to action if they realize they're suspected. If they have any corp roles or access, your first action should be to strip them - not talk to them, not try and sort things out. If they don't, sack them. Block access to every corp channel and forum that is not automated on corp membership.
3) Keep a bulwark for rebuilding if your corp is hit by a thief, and keep it separate to corp funds and assets. Keep valuable and portable corp assets in personal, not corporate hangars. Split your reserve fund, if possible, over characters on different accounts with different log-ons and different passwords. And ofc, change your passwords frequently, including passwords to locked containers in corp hangars. Keep good records of your assets and check them frequently. You can cross reference pilot in-pod activity with asset disappearances.
4) Accept that you will be infiltrated or betrayed at some time during your corporation's lifetime. You are not the smartest person in EVE, and someone who is smarter than you may very well fool you, rip you off, take your stuff, and gloat about it. It's like getting suicide ganked - it's a PITA, but it shouldn't make you emoraegquit.
5) Some other stuff that I'm going to keep to myself for now ...