MOST second stages fail open. Like DD said, Poseidon regs are known to fail closed - although some of the most recent models may not.
With first stages, it depends on the problem... But most problems that I know of will cause a first to fail closed.
That's why simple is good... The simpler, the better.
There's several ways to combat these problems:
1. Have your regs serviced regularly. Of course, this means no guarantees... In fact, most reg failures that I've known of have happened soon after a service.
Nonetheless, servicing and preventative maintenance can go a long way to keeping your regs functioning.
2. Don't dive solo. Have a buddy that's skilled and dependable so that you have a secondary source of gas. Those that dive solo often use a pony or a SpareAir, but those units have a reputation for being less than dependable. Opinions vary, so do your research and make your own choice.
3. Dive with a manifolded pair of doubles whenever you're in a situation where an ESA is impossible. A manifolded set of doubles gives you two second stages, two first stages, two bottles of gas, and the ability to switch, on the fly, to use whatever is working at the time. For example, a failed first stage would not prevent you from using the other... And still be able to allow you access to all of the gas available.
4. Don't forget that your BC inflator can supply you with gas in a pinch... That is, in a way, you actually have three second stages.
Hope that helps.