At the very beginning of the story, Dr. Watson is describing for the reader Sherlock Holmes's relationship with a woman called Irene Adler. Dr. Watson says that as far as Holmes is concerned, Irene Adler "eclipses and predominates the whole of her sex," meaning that to Holmes's mind, Irene Adler is the most impressive woman he has ever met. Dr. Watson also clarifies, however, that Holmes did not love Irene Adler, and he says that this is because "all emotions … were abhorrent to [Holmes's] cold, precise but admirably balanced mind."
From this quote, it seems clear that Dr. Watson is not saying that Holmes has a balanced mind in the sense that there is a balance between intelligence and emotions. What Dr. Watson means is that Holmes's mind is finely balanced, or calibrated, like an efficient machine. Indeed, shortly after Dr. Watson describes Holmes's mind as "balanced," he also calls the same mind "the most perfect reasoning and observing machine" that he or anybody else has ever seen. Holmes does not love Irene Adler because emotions are too unpredictable, and would thus upset the carefully calibrated balance of his mind.
In the Sherlock Holmes stories, the defining characteristic of the eponymous Holmes is an ability to balance and weigh up different clues, and to arrive, by a process of deductive reasoning, at solutions that nobody else can see. In this sense, he is, as Dr. Watson says, machine-like. In modern terms, his mind is like a high-functioning computer, into which complex data is entered and out of which beautifully logical answers emerge.