Warning! Spoilers ahead for The Batman
As The Batman takes place early on in his vigilante career, there are very few people who know Bruce Wayne roams the streets at night seeking justice and vengeance as the Batman, especially when compared to the number of characters who know his secret identity in the comics. That being said, future spin-offs and sequels like The Batman 2 could certainly increase that number over time as more characters both good and bad are introduced. Here's everyone who knows Bruce Wayne is Gotham's resident crimefighter (in The Batman as well as in the comics).
Naturally, Bruce Wayne's loyal butler has always known about his nocturnal activities as Batman, and he even helps Bruce solve the Riddler's codes and ciphers in Matt Reeves' The Batman. Having known Bruce the longest, having cared for the young boy after Thomas and Martha Wayne died, Alfred taught Bruce to fight and defend himself while admitting that he wasn't fully equipped to be the ideal father figure Bruce needed. Regardless, Bruce still loves Alfred as the only family he has left. Without Alfred, there's no Batman (and the same holds true in the comics as well).
Related: The Batman's Arkham Twist Breaks The Wayne Family Legacy
Following Batman's discovery that Bruce Wayne was one of the Riddler's next targets in The Batman, he attempted to race back to Wayne Tower in the Batmobile, fearing for Alfred's life. However, the damage had already been done with Alfred being caught in an explosion caused by a bomb sent by the Riddler. While Batman tried to call Alfred, the Tower's housekeeper Dory answered the phone to let Bruce know that Alfred had already been hospitalized. While it's unclear how much or how little Dory knows about Bruce
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