The protagonist is, as with most of Arthur Conan Doyle's stories, the illustrious Sherlock Holmes.
In this story Holmes faces the formidable Dr Roylott - ably assisted by his great friend Dr Watson. The case is brought to Holmes by Helen Stoner, a delicate young lady fearing for her life after the violent and inexplicable death of her sister. She is suspicious of her volatile stepfather as he would have been significantly in debt should Helen's sister Julia have married and claimed her inheritance.
With the usual twists and red herrings (a ‘band’ of gypsies in this case) Holmes the protagonist manages to thwart Roylott the deadly antagonist by turning his own evil snakebite plot on himself.
A case could be made that it is Dr. Roylott who is the protagonist in "The Adventure of the Speckled Band." That would make Sherlock Holmes the antagonist in the drama. Here is a definition of the term "protagonist" by Lajos Egri in his invaluable book The Art of Dramatic Writing.
The pivotal character is the protagonist. According to Webster’s dictionary, the protagonist is “one who takes the lead in any movement or cause.”
Anyone who opposes the protagonist is an opponent or antagonist.
As we see, a pivotal character never becomes a pivotal character because he wants to. He is really forced by circumstances within him and outside of him to become what he is.
The protagonist is the character who starts the action, who makes the story happen, and who is mainly responsible for its continuation. Dr. Roylott started the whole story two years earlier when he killed Julia. We do not know that for sure, but it is not necessary to know the identity of a protagonist immediately or to know his motive. For example, there are many movies and television shows in which someone is committing a series of murders. No one knows who he is or why he is doing it, but that person is the protagonist and the investigator's problem is to expose him. As another example, in "The Hound of the Baskervilles" it is Mr. Stapleton who owns the hound and plans to murder Sir Henry Baskerville, but nobody knows that until near the end of the long story.
If Dr. Roylott had only murdered Julia and had never given Helen cause to fear for her own life, then Dr. Roylott would not be the protagonist. But he now wants to kill Helen for the same reason he killed Julia. Helen wants to get married and Roylott will be forced to pay her one-third of his annual income. According to Egri, "a pivotal character never becomes a pivotal character because he wants to. He is really forced by circumstances within him and outside of him to become what he is." Dr. Roylott is being forced to try to murder Helen because Helen is engaged to be married. His income has already shrunk drastically. His big house is heavily mortgaged. He would be ruined if he had to pay Helen 250 pounds a year out of his total annual income of 750 pounds. He murdered Julia for the same reason he intends to murder Helen.
Arthur Conan Doyle introduces Dr. Roylott early in the story by having this violent man come directly to Baker Street and threaten Sherlock Holmes. His intrusion and his threats of bodily harm are proof of his strong motivation. He undoubtedly knows that Holmes suspects him of killing Julia as well as of having sinister plans against Helen, but Roylott is half insane and doesn't care what Holmes thinks as long as he can't prove it. And he hopes to keep Holmes from proving it by trying to frighten the detective into dropping out of the investigation and staying away from Stoke Moran. Holmes is a worthy antagonist to this protagonist. Holmes is working for Helen Stoner for nothing, but that doesn't stop him from pursuing his investigation until the problem created by Dr. Roylott is solved.