There’s Batman, there’s the Guardian, there’s Green Arrow, Hawkeye, and (sort of) Captain America. Also the Falcon and the Manhunter, I believe. In fact I’m sure there’s a lot of them, though only because there are so many superheroes in general. My guess is that the nonpowered heroes tended to crop up in the ’40s and became less commonplace during the Silver Age and after, though the examples above show that they didn’t die out.
This is leaving aside all the Batman satellites. In the interest of franchise homogeneity, anyone who wants to be Batted cannot have special powers. Though, given the number of people who keep getting superpowers, and the number of people who want to be Batman’s crimefighting associate, and the difficulty of Batman’s line of work, you’d think there would be at least some people with superpowers who’d be trying out for the role of girl-Robin or the Silver Bat or whatever. (My thoughts on Kathy Kane, and commenters’ very informative remarks about Bat-training, can be found here.)
I don’t count Iron Man as nonpowered, though he couldn’t have got into the Legion of Superheroes: they think gizmos are cheating. Of course, where you draw the line between Batman’s ton of gimmicks and Tony Stark’s armor could be the subject of much debate. But I’m sure the line’s there someplace.