It might go some way towards lowering the popularity of vampires in the popular consciousness, but IMHO it's not enough. It was Polidori's "The Vampyre" (a parody of his boss Lord Byron) that transformed the vampire from a bloodsucking version of the mindless zombies that filled medieval folktales into characters with human intelligence and minds. This made it possible to make vampires both interesting villains and sympathetic protagonists and anti-heroes, which allowed the 19th century boom in vampire stories that continues to inspire authors.
If you wish to free us of a plague of vampires, you must kill off either Byron or Polidori. Or maybe fix Byron's brain chemistry (perhaps he never tries opium?) so that he spends his life writing about puppies and rainbows and doesn't inspire monster stories.
Eh...I could just as well stop that evening in Geneva and strangle both the modern vampire and Frankenstein's monster in the cradle.
Still, my proposed POD (while not actually getting rid of them) is going to alter the popular perception of the vampire a fair bit.
Without
Dracula the main 19th century source material for vampires is going to be Polidori's aforementioned short story and Rymer's
Varney the Vampire. While these stories established much of the modern vampire archetype (sexualised aristocratic blood-sucker) the loss of Stoker's work is going to have some important key effects.
Firstly, vampires are going to be British by default. It was Stoker who succeeded in taking them back to the Balkans, whereas before (as you pointed out) they were ultimately supernatural Byron proxies. The image of the vampire could well become tied to that of the landed aristocracy, which has interesting possibilities further down the line.
Secondly, they are going to have far fewer weaknesses. Stoker was the first modern author to make use of things such as garlic and crucifixes as weapons against the vampiric menace. Rymer's Varney character could even walk in sunlight unpertubed. Creative writers will undoubtedly find some way to kill the villain off, but it is hard to say if they will seize upon the same ideas as Stoker or take it in a whole new direction.
Finally, vampirism will be percieved far more as a suffering affliction than a malevolent presence. Rymer's Varney was cursed with vampirism and actively loathed his condition, ultimately casting himself to a volcano to end his suffering. Furthermore, between fits of vampirism he seemed and behaved to be a normal human being rather than a member of the undead. Vampires may become more akin to werewolves in popular perception, to be pitied as well as feared for their unending suffering at their affliction without any true hope of redemption.