And in his earliest appearances, Batman had no qualms about killing villains or letting villains die. He even carried a gun. After all, he was based on pulp antiheroes such as The Shadow and The Spider.
That's something that tends to be hugely overstated. For one thing, it was only true in the first few months of the character's existence; his strict no-killing policy was handed down by the editor before the character was even a year old. (Some people claim that the change didn't happen until the Comics Code Authority was established a decade and a half later, which is ridiculously wrong. Comics as early as 1940 were stating emphatically that the Batman never killed.) It's also untrue to say he had no qualms about it in those early issues. In the issue that led to the no-guns crackdown, Batman shot up a truck full of superstrong monsters (and at least one human driver) from the Batplane, but said, "Much as I hate to take human life, I'm afraid this time it's necessary!" And the only previous time he'd been shown shooting anyone was to kill vampires with silver bullets; otherwise, he'd just been shown firing at inanimate objects or brandishing guns in cover or splash-page art that wasn't part of an actual story. Although in his fourth issue, he did break the villain's neck with a kick.
As to the question of double standards about killing... Audiences assume that the villain is a bad person killing good people, and therefore it's an act of heroism and justice to end the villain's life.
And as I've said, I think that's a bizarre and self-contradictory notion. If the villains are bad because they kill, why is it good for heroes to kill? That just doesn't make sense. The idea that "If they do something bad to me, that makes it okay for me to do the same bad thing to them" is a childish excuse for bad behavior. Other people's misdeeds are not a license to sink to their level. On the contrary, they're a reminder of the importance of being better than that.