A democracy doesn't mean that everybody gets to do or not do whatever they want - that is what is called anarchy.
As for voting, is voting a right, a privilege, or a responsibility in a democracy? I think a democracy would run better if it tended more towards being a responsibility.
Maybe if more people participated there would be more people invested in the outcomes. Maybe if more people voted, politicians would change their behavior to appeal to a genuine 50%+ rather than just 50% of 37%.
I think it is an idea worth considering. As you said, many people are fed up with the whole political process - so surely we would not want to keep doing the same old thing?