Demonstrations are inconvenient for Athenians and visitors, but mostly just provide extra work for the police. As outsiders, begin to take the situation seriously if the demonstrations morph into terrorism and civil war.
The DrsC were in Greece just over a month ago and, as persons who didn't speak or read Greek, could not visually determine anything was wrong. We saw no military deployed on the streets of Athens, no sandbagged doorways, no barricades, no armored vehicles, none of the usual signs of a government under siege. Everything looked as normal as a relatively modern foreign city ever looks - busy streets, modern airport, well-dressed people, graffiti.
As long as the Greeks continue to trust their electoral process to produce governments which reflect their feelings, things are okay. Once you see bombings and the like, the emergence of a credible armed force in opposition to the government, that is when things can spiral into chaos, then worry.