The short answer would be "it depends", but considering where you posted this, I have a feeling you are not looking for short answers.
First lets determine what an extremist is. It seems that the term is used to refer to individuals whom the public at large places outside of the overton window, and thus are necessarily unacceptable and detrimental to those who are associated with them. In the strict sense of the word, it means a person or group who takes an ideology and takes it to the extreme, usually by turning generalities into absolutes and refusing to interpret situations through other perspectives. Extremists usually are selective in the parts of the ideology they stick to, so any given movement can have different sets of extremists that act in opposition to each other. For example: a group of Christian extremists may bomb abortion clinics and attack homosexuals, while another may stand for non-violence with no exceptions and absolute charity, giving all of their possessions to those in need. One represents what the public sees as the bad side of the movement, while the other represents the positive side.
The first factor to be taken into consideration is how much publicity the extremist faction gets in comparison with the moderates. In the case of the Muslim extremists, it is obvious that they are receiving more attention than their moderate counterparts, since the term "Muslim" is most often associated with the terms "terror", "terrorist", "attack" and "extremist". In this case, the impact of the extremists in the West is greater than that of the moderates, despite the huge numerical advantage of the latter, cementing this association.
The matter of time is also relevant. If an extremist faction appears too early in a movement's history, that movement is much more likely to be associated with it. The longer a movement has existed before the extremists made themselves known to the outsiders, the better for the movement.
It is also possible for the presence of the extremists to serve as an image booster for the moderates when they dramatically reject them during a schism. If the moderates stand with the public opinion and make it very visible how different they are from the extremists (with an enthusiasm even grater than that of the masses condemning them), this can be a good marketing strategy.
The examples above assume that the extremists are of the bad kind, which seems to be the kind that is either more prevalent or that receives the most publicity. I can think of one situation where positive extremists can help the movement, and that is by serving as poster boys and representatives in general (somewhat like missionaries), so that the first impression the masses have of the movement is a positive one. What is important here is for the moderates to explain that not everyone who joins the movement needs to sacrifice as much as the extremists, or believe as fervently as they do. Note that despite the fact that this is the only scenario I can think of for "positive" extremists, it is very general and useful.
There is probably much more than can be said about this topic, but I'm afraid that's all I've got right now.