As mentioned, Batman's killing diminished sharply in Detective Comics after Robin's first appearance in April of 1940.
That same month began Batman comics, and especially in the beginning Batman's adversaries regularly ended up dead. The main difference being that Batman related deaths were now often due to less direct means. Deaths in Batman comics generally happen one of two ways: someone else shoots the antagonist during a fight with Batman or the antagonist falls to their death during a fight with Batman.
Some early Batman related deaths in Batman comics:
Batman #1: The second story has the only killings in the first issue, and is arguably the most violent pre WWII material. (This is why I feel it should be considered the last Pre-Robin 'original' Batman story) In it he shoots people from his plane, and gruesomely hangs one of the 'monstermen' from a noose, amongst other killing. Batman's quote: "As much as I hate to take human life, this time it's necessary." Batman ends up leading to seven deaths if you count the mentally ill people that Dr. Hugo Strange has turned into 'monstermen.'
Batman #2: In the second story Batman punches the antagonist (who has stopped attacking at this point) down a flight of stairs, where he breaks his neck and dies.
In the final story "The Case of the Missing Link" Batman fights with pygmies on the roof of the train and four are struck by a low bridge, later Batman & Robin cooperate to knock a giant caveman to his death off a high scaffold. Would being struck by a bridge while standing on the roof of a train moving at full speed kill a pygmy? Is this a question fit for Monty Python? Yes and yes.
After these two issues, things change. Deaths in Batman comics gradually decrease in frequency, by 1945 there would be maybe one person indirectly getting killed due to Batman's involvement every 4 or 5 issues. This means approximately one indirectly caused death per 200-250 pages of material. In contrast, Batman very clearly killed eight people in the original 110 pages of Detective Comics at the beginning in in 1939, or approximately one death every 13.75 pages.