FBinNY 

To start with, this is a hypothetical question about something extremely unlikely to happen on a two brake bicycle.

The systems are independent, and even if not in the best condition, the odds against of catastrophic failure at the same time are staggering. Yes, one can fail, and often will, but that gives you plenty of time to bring the bike under control and consider your options relating to continuing with only one brake, ie. walk down steep hills, pick up and carry a rock to use as a brake shoe on the front tire, or just hone for a ride.

If riding with only one brake which fails, the shoe on tire behind the fork may be one of the poorer options. It can compromise steering, and has a high risk of causing an endo if not modulated well. Also consider the reality that the action of the wheel would be to increase braking, so the likelihood of rider error is increased.

A better option is putting the left foot against the rear tire at the chainstay, Here the worst that can happen is locking the wheel and dumping to the side. This will likely cause road rash, and possibly some other injuries, but side falls are generally safer than endos.

Other options include riding it out using posture to maximize wind drag, looking for a soft landing spot, looking for a potential turnoff toward the uphill side, etc.

So, IMO it's best and easy enough to prevent this scenario altogether, but ride smart once you find yourself down to one brake, being ready to deal with it if that one fails too.

BTW- if we look at causes of injury on bikes,and consider the level of risk we assume every day by riding in the first place, the risk if dual brake failure is so low that it wouldn't even amount to a statistical blip.