The '89 Batman franchise was mostly based on the original comics from the '30s and '40s where he killed criminals all the time, sometimes with an actual revolver.
His moral stance on killing is actually a significant and underappreciated arc throughout the three films. In Batman, he doesn't kill anyone and very much tries to save Jack Napier from a fatal fall in the chemical factory. Then, when he discovers it was the Joker who murdered his parents, he pulls out all the stops, fully intends to kill him and along the way ends up killing half his henchmen as well.
In the ultra-dark Batman Returns, he is still in this place. He kills numerous henchmen, but is eventually stunned into putting all this in perspective when he sees the parallels between himself and the psychotic Catwoman, whom he's falling in love with. He tries to "save" her at the climax by talking her out of it all, but fails.
In Batman Forever, there's heavy focus placed on this in his dealings with Robin and his revenge-plans for Twoface, which goes into detail about how it will just leave him without a course and looking for another and another to kill until one day he's empty. He finally kills Twoface himself. His final kill. This one, however, does give him satisfaction, not because it ended Twoface's life, but because it saved Robin's from the same path.
I actually wrote an award-winning essay on this in high school, maybe the best assignment ever.
