Everything has to do with the historical context of the novel. The Strange Case of Dr. Jekyll and Mr. Hyde was published in 1885, which marks one of the most puritanical times in English social history.
First, the influential and powerful 19th century was coming to an end. The 20th century was on the horizon, and many 19th century Victorians though that it would be safer to stick to the "old school" paradigms and make them even stronger in order to avoid a new order to bring the righteous Victorian rule to chaos.
Therefore, the social media of the time (newspapers, flyers, articles), and political powers developed a pro-morality and pro-puritanism movement in the 1880's that was based on accusing everyone or anyone who does something outside of the norm as a "criminal".
For example, the Criminal Amendment Act was enforced that year aiming to deflect a growing tendency in homosexual practices, particularly among members of the upper classes with lower class escorts. Since the English were so exacting at differentiating people based on origin, birth, family name, profession, income, and claim, what would be easier than to blame the lower classes for inciting the upper classes to commit heinous crimes?
Therefore, based on the historical context of the novel, we can conclude that it is more effective to have all upper class characters because, as they commit less than respectable acts, the audience realizes that crime is not precluded by social status; it is nature, perhaps, and not nurture. In not so many words, it is more effective to have all main characters represent the upper classes because that goes to show that even those who are well-bred and highly educated are capable of acting lowly, of having cheap morals, and of committing crimes.