What can a passenger do if (s)he objects to the flight route? Say that after a passenger learns of or views the flight plan, she opposes it (eg due to safety).

She could just refuse to board the flight and forfeit her ticket and costs, but are there less costly and dire and radical options? Despite being only a passenger, could she explain to the flight crew (plausibly via the cabin crew?) her reasons and thus request the reconsideration of the flight route? 

Supplementary: I elucidate that I do NOT mean for her to compel or impress a specific route (replete with all the waypoints, VORs, etc...) upon the flight crew, but only a general concern for avoidance. 

As an instantiation, if I were a passenger and had apprehended my flight to overfly Ukraine, even outside of the no-fly zone, then I'd remonstrate against it, in favor of a more northern one that would substantively avoid Ukraine.
The passenger can in all likelihood do nothing that is going to change the route.

As you point out, there is always the option to not board. 

On long range flights like this, the routing will (depending on winds aloft and other conditions) almost certainly require a fairly direct route to insure that additional fuel stops are not needed. In cases where routes go over warzones or other dangerous areas, they are often rerouted, but in the case of Ukraine the conflict has until now been isolated and military in nature.

Either you trust the flight crew to also be aware of these risks, or you don't. Get on board, or not. I'm sure you could raise the issue with a gate agent, and if you made enough noise or escalated it, you might be able to get a refund (note: I have no idea). However, you also might be perceived as the source of the risk and detained!