I don't think the citizens of Earth Prime-Time are aware that Batman was really Bruce Wayne; but there could be some references scattered about in various series. I suppose there are plenty where it's mentioned that Adam West played both roles. (Zonks from the Tooniverse don't count, so I'm safe as far as 'Family Guy' goes.)
But the fact that Batman operated out of his secret headquarters known as "The Batcave" would be common knowledge. Villains were always trying to gain access to it and newspaper accounts eventually would have reported this.
While I was on vacation, I saw the Batcave mentioned on two very different series, which didn't have anything to do with superheroes. First up was an old episode of 'Law & Order: Criminal Intent' in which Detective Robert Goren compared an old warehouse used as a murder site to the Batcave. The other reference came from one of the last episodes of 'Surviving Suburbia' to be shown, in which Bob Saget's character thought of his son's closet as his personal Batcave, where he could escape from his wife.