Batman has been referenced in three of the universes
Earth-1 (main universe)
Bruce Wayne exists in this universe, is will known, and lives in Gotham. There is no confirmation of him engaging in vigilante activities.
Oliver Queen: Photos can be doctored. They could have put Bruce Wayne's head on that body. Has Bruce Wayne left Gotham to hang out in Star City recently? No.
Arrow season 6 episode 2, "Tribute"
Additionally, the future newspaper shown in the Flash mentioned a "Wayne Tech" company, which may possibly be associated with Bruce Wayne.
Flash season 1, episode 1, "Pilot"
Edit (December 11, 2018): The fact that Bruce Wayne is Batman in Earth-1 has been confirmed during this year's crossover when Supergirl, Green Arrow and the Flash visit Gotham City. Precisely, he was Batman until he disappeared from Gotham City 3 years ago. His cousin is now replacing him as Batwoman, and will have her own spin-off show in 2019, so we may have more information later on. [Edit written by Taladris on ibid's suggestion]
Earth-2 (Flash alternate universe)
A character by the name of "Bruce" was found on Earth-2's Barry Allen's phone alongside "Hal" and "Diana". Nothing else was said about him. (Note that Earth-2 Barry doesn't have the speedforce.)
(source: wordpress.com)
Flash season 2, episode 13, "Welcome to Earth-2"
Earth-38 (Supergirl universe)
An unnamed vigilante matching Batman's description was said to have once worked with Superman.
Kara: My cousin worked with a vigilante once, tons of gadgets, lots of demons. Vigilantes are nuts!.
Supergirl season 2, episode 7, "The Darkest Place"
James Olsen: So, people see this big guy in this scary mask and they freak out, they run away.
Winn Schott: Just like Clark's friend.
James Olsen: Oh, you mean...
Winn Schott: I feel like they're more frenemies
Supergirl season 2, episode 20, "City of Lost Children"
