It would be tempting to just run all your key tasks on a secure computer that's never connected to the Internet. In fact, when we recently spoke to MYOB's CTO Simon Raik-Allen he suggested that many small business customers of MYOB do exactly that. Their accounts run on a computer that is never connected to the network.
In the case of Mat Honan, his online details were compromised when someone called Apple and fooled them into resetting Honan's password thus giving them access to his iCloud account. From there, they used that email address to reset passwords on all of Honan's other services thus locking him out of his own data and taking over his online identity. Apple has since altered its procedures to reduce the risk of such an action happening again.
So - what can you do about this? Like any system or process try to avoid having everything interconnected in such a way that failure of any one part brings the whole thing down.
In Honan's case the problem was that once his iCloud was compromised, it provided access to all his other accounts as they were interconnected. Try to keep your online accounts isolated from each other.
Many banks now offer a new feature called two-factor authentication. Instead of only relying on a single username and password combination, they use a second identifier. Typically, this is a device with a passcode that changes every 60 seconds. That passcode is synchronised with a service that is very, very difficult to hack. Aside from banks, Google offers two-factor authentication (using a different mechanism) as do other online service providers. They are a little more annoying but worth considering.
Finally, back-up your important information so that if systems are compromised and data is damaged you have a method of recovery.