1962:
In the DCAU there is no mention of Bruce Wayne's true age. But we know that Bruce's parents were killed leaving a showing of the adventure film "The Mark of Zorro" (per JLU episode "For the Man Who Has Everything").
The animated Batman was inspired and shares some continuity with the Tim Burton Batman films, who's look was inspired by graphic novel mini-series The Dark Knight Returns by Frank Miller. In that story the movie that Bruce's family sees is the 1940 "The Mark of Zorro" with Tyrone Power, which was released in November of that year. I decided to keep that homage intact, but to keep Bruce's age in check I theorized that Bruce's parents took him to see the 30th Anniversary of the film in November of 1970.
It has been alluded in several episodes that Bruce was 8 years old when his parents were murdered.
Justice League episode "A Better World, Part II": "...we've made a world where no 8 year old boy will ever lose his parents because of some punk with a gun." - Justice Lord Batman (to Justice League Batman)
JLU episode "Kid Stuff": "I haven't been a kid since I was 8 years old." - Bruce Wayne
JLU episode "Epilogue": "Stop me if you've heard it before. You're 8 years old. Your parents have just taken you to a rousing adventure film; a grand time is had by all. But unknown to you a mysterious figure hides in the shadows. My plan was simple: the killer would leap out at you and kill your family. The trauma would put you on the path to becoming Batman." - Amanda Waller
If Bruce is 8 years old when he sees the 30th Anniversary release of "The Mark of Zorro" in November 1970, then it follows that Bruce would most likely have been born in 1962; 8 years before those events.