Sir Charles Baskerville believed the legend of the Hound of Baskerville, and it lead to his death. When he saw a hound coming his way, he thought the legend was true and died of fright.
Holmes does not believe the story. In fact, when Dr. Mortimer brings it to him he says it is interesting only “to a collector of fairy tales” (p. 11). It is facts that Holmes wants, and he does take the case because he wants to find out who is exploiting the legend and how, and of course why.
Sir Henry does not completely believe the legend, and he takes up his home in the moor. However, he is nervous and worried because he thinks someone is trying to kill him like Sir Charles.
Watson does not believe the story, but Stapleton pretends to. Stapleton of course needs to urge the story along.
Stapleton may fall in with such a superstition, and Mortimer also; but if I have one quality upon earth it is common-sense, and nothing will persuade me to believe in such a thing. (p. 69).
More importantly, Watson believes that believing in the legend is beneath him, and he would be letting Holmes down if he believed it.
To do so would be to descend to the level of these poor peasants, who are not content with a mere fiend dog, but must needs describe him with hell-fire shooting from his mouth and eyes. Holmes would not listen to such fancies, and I am his agent. (p. 69)
Of course, the locals seem to view the legend as just a legend. They believe it as people would any ghost story, as a diversion.