What can I do as a bystander on the ground if I see an aircraft above me that appears to be failing?
Honestly, very little.

First, be clear as to whether you are actually see something failing. If you see flames, or a wing has broken off, then yes, it's a problem. However, if you see someone doing a slow spiral towards an empty field with the power idling, they're more likely to be practicing engine out procedures (something that's required for all US pilots to practice and demonstrate during their checkride for their certificate). 

In the event that you actually see an impact, fire, off-airport landing and so on, call 911 and give them your location. They'll send the appropriate crews. 

Otherwise, if the aircraft is still flying, there's a strong chance the pilot(s) are already talking to ATC about their problem, if it actually exists. There's nothing you really can or need to do in this case.

Quick Anecdote: I've been told the story of a local police department who gets phone calls virtually every time a citizen lands his small airplane in his private field. Fire/police/medics/etc are summoned, when absolutely nothing bad has happened. This is why you should make an attempt to verify that a problem actually exists before reporting it.