The protagonist in this one is a doctor, who has been seeing to the needs of a family with an infant child. Misfortune starts befalling the family as multiple accidents and one or more fatalities occur in the home. I can't remember if there are multiple murders, or just multiple accidents eventually leading to the death on one person (the mother, IIRC). I do remember something about a child's toy left at the head of the stairs, leading to a terrible fall. I believe the mother is killed when a gas lamp in the home is snuffed out in the bedroom, leading to her asphyxiation/poisoning from gas fumes.
As the story unfolds, the doctor becomes suspicious of the events, and eventually comes to the horrifying realization the the infant is behind the killing(s). It left the toy at the head of the stairs, somehow crawled over to the gas lamp and snuffed the flame out, etc. I believe the doctor reasoned that the child was born aware, and was resentful of having to leave the womb.
The story closes with the doctor resolving to stop the child. In the closing paragraph, he takes a scalpel out of his medical bag. As he approaches the child to slay him, he twirls in back and forth, catching the infant's interest with the flashing on the blade in the light.
I can't really narrow down when the story was published. I believe I read it in the 80s, in an old sci-fi anthology paperback I picked up at a used book store. I think the story collection was an old one even then. As for the story's setting, the presence of a gaslight in the home makes me think early 20th century or even late 19th century.