There seem to be two main reasons for banning the party in many people's minds:
1) Many of the ideas presented by the party are very scary in the eyes of the Greek population. Such propositions include a massive overhaul and control of the country's immigration system in an attempt to re-establish a "pure Athens". Such motives have been compared to Nazi standards and desires in the past, although Golden Dawn denies any connection or inspiration from Nazi ideals.
2) Although Golden Dawn functions as a political party, it also acts as a social movement which many people have attached themselves to. Followers of Golden Dawn ideology have been known to enact violence against those known to disagree with their ideologies. In fact, the recent murder of several socialist activists is what sparked the debate currently.
I am conflicted on how to feel about this situation. On the one hand, I find accepting the prohibition of ideas or public display of ideology to be difficult. Banning something for being too fascist seems somewhat ironic to me. On the other hand, many of the ideas promoted by the group may be deemed too dangerous to be accepted in civilized society; the idea of a potential holocaust of immigrant populations, especially the Turks, is very scary to me. I would also say that Golden Dawn has crossed the line from being "political" to being dangerous as it becomes to act more and more like a gang.
In the end, I cannot come up with a positive claim of what action is best. However, at the very least the government of Greece should take greater steps towards preventing these violent acts from acting in the future. Also, if the political party could be proven to have direct connections to any mass murders, it could be punished and stopped that way.