What can a passenger do if (s)he objects to the flight route?
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!