He would most likely feel bad about it for a while but for the most part, his life wouldn't change that much. Joker appears very rarely as Batman's foe, so killing him wouldn't change that much. Batman has actually tried to kill Joker once in canon. That happened in New 52's Endgame storyline.
But to his shock, he doesn't find a heartbroken and disheartened crowd. Instead, he finds people smiling and laughing at his memory, inspired by his actions. Everyone decides to do what Batman did, and turn tragedy into a chance to be better. Having lost his nemesis, Joker no longer has a true purpose in life.
In fact, without Batman, the Joker has no motivation to resume his twisted ways. Following their last fight, Joker goes on to lead a mundane life — eventually leaving the crime world altogether to work at the Gotham Department of Motor Vehicles.
Batman would also feel the same if Joker were taken down while trying to kill others. If he were killed out of revenge, Batman would feel an obligation to bring the killer in, though it might not be popular among the general population.
The true reason why Batman doesn't kill the Joker has been revealed. The Dark Knight believes that there is still some part of the Joker that is worth saving. This belief in humanity and finding it even within Gotham's most depraved criminal proves that Batman is and always will be one of DC's most caring heroes.
Batman's “no killing” rule is to ensure he always acts in service of justice, not vengeance, and that he never let himself become so above and beyond the law that the last restraint on his actions is removed.
1 Ben Affleck's Batman Killed Over 25 Criminals With Lethal Weaponry And Brutal Tactics.
Joker even managed to poison Batman with his blood forcing the Dark Knight into helping him get the cure for the disease.
Wayne is replaced as Batman by an apprentice named Jean-Paul Valley (a.k.a. Azrael), who becomes increasingly violent and unstable, tarnishing Batman's reputation.
Batman manages to get Robin out of the warehouse, but is killed in the explosion. With his dying words, Bruce attempts to convince Jason not to take revenge on the Joker and to be strong for his family. Jason, Alfred, and Barbara Gordon put Bruce to rest next to his parents, and Dick Grayson succeeds him as Batman.
Batman is convinced that if he kills the Joker he'll sink to his level and become something like him. Joker echoes this sentiment frequently while badgering Batman about their many similarities. If Batman really believed he was as limitless as he says he is, he wouldn't be so scared of losing himself so easily.
Because he's valuable intellectual property, so the rules of the universe as set by the DC writers and editors don't let him be executed.
Why does Batman keep the Joker alive? Death of the Family has a new reason: Batman confides to Alfred that the main reason he refuses to kill Joker is because he sincerely believes killing Joker wouldn't make things any better. Gotham would just send someone worse, or bring Joker back from the dead, or something.
How is Batman infected with Joker blood? He took the cure in Arkham City, prepared by Freeze. That's how Joker died, because Bats took the cure, saved a bit for Joker but Joker stabbed him in the back and he dropped the vial, breaking it.
Tim Burton's first Batman film released in 1989 contained the shocking revelation that a young Jack Napier killed the Waynes. Fans of the film know that Jack Napier would later become The Joker, adding a whole new layer to the infamous Batman/Joker rivalry.
In Batman's origin story, Joe Chill is the mugger who murders young Bruce Wayne's parents, Dr. Thomas Wayne and Martha Wayne. The murder traumatizes Bruce, inspiring his vow to avenge their deaths by fighting crime in Gotham City as the vigilante Batman.
Alfred Pennyworth, after a varied career, was employed as the Wayne family valet when Bruce Wayne's parents were killed. Alfred raised the young orphan and reluctantly aided him in his quest to become Batman.
Superman & Batman: Generations
In the process two souls enter and the Pit destroys one while imbuing the other with youth and immortality. Batman survives this process, subsequently becoming immortal (Although he is not a true immortal, as he ages one year for every century).
After capturing Scarecrow and returning to Panessa Studios, the main floor between the cells was painted with a large, red logo of Red Hood. It was Jason's way of showing Batman that he was back and that he had forgiven him.
The condition known as pseudobulbar affect (PBA) is characterized by brief uncontrollable outbursts of crying or laughter that are incongruent with the patient's feelings of sadness or joy.
While in Arkham City, Batman was ambushed by Clayface (disguised as Joker) and knocked out by Harley Quinn. After he woke up, Batman discovered that Joker suffered from a horrible side effect of Titan which gave him a fatal disease that progressively killed him.
The Batman Who Laughs is arguably the most dangerous version of Bruce Wayne that has ever lived. After finally being pushed over the edge, Batman kills the Joker. However, in doing so, Bruce is exposed to a special form of gas that steadily transforms him into the Joker.
After discovering their secret lair, Batman becomes lost in a maze, finding himself hunted by a Talon, a very deadly assassin for the Court. After being tortured both physically and mentally, a Talon is ordered to deliver the final blow, ultimately killing Batman in a very violent and bloody way.
The page everyone remembers from Batman: Knightfall is Bane “breaking the Bat” over his knee. But Bane beat Batman by working smarter, not harder. Bane was able to defeat the Dark Knight physically by first exhausting him spiritually (and, well, also physically).