I don't think it was Batman per se, but something definitely did happen. I wonder if it was just the end of a cycle for a certain type of action film, like how Westerns died out in the 70s-80s to a large extent.
Action films haven't faded, though they have changed. After the end of the Cold War, there was no massive, easily identifiable enemy so gone was that threat. Perhaps there was no more need for musclebound, killing machines since the threat no longer seemed to necessitate it. The threats became more nebulous and action films changed to reflect that. I can see how Aliens, Lethal Weapon, and Die Hard punctured holes in 'traditional' action films, but I would throw in the Indiana Jones films and other Harrison Ford fare and some of the movies like Romancing the Stone that were like forerunners of the rom-com movies today. They showed a different type of action star and were profitable, setting the stage for the eventual decline of the old school action stars. Though thinking about it more, not all the action stars even during the 80s were musclebound. Chuck Norris had a pretty lean physique from what I can recall. As did guys like Michael Dudikoff from the American Ninja movies, though Norris at least had the required badassedness, with Dudikoff I can't remember.
The soldier was replaced by the cop, largely, as the action star of the 90s IMO, and you didn't need a he-man to play a cop. Also there has been a change in how masculinity has been viewed over the last 20 or 30 years and I wonder if that hasn't had an impact on what we accept as action stars today. Also, more became required of action stars than to have big muscles and well-timed quips. The action stars of the 90s: Wesley Snipes, Nicholas Cage, Keanu Reeves, Bruce Willis, Jet Li, Chow Yun Fat, Jackie Chan, etc. didn't have massive physiques but usually they were better actors, or in Chan's case, could pull off comedy and action more convincingly than Arnold or Sly. Even Seagal and Van Damme, who I would consider more in the Arnold and Sly mold, weren't as big as them physically.
For good or ill, the 90s opened the gates to all types of actors in action films and to some extent, the action genre got pushed to the straight to video market and there stopped being viable 'action' stars, and by that I mean actors whose bread and butter films are action films. Until today, you've got Will Smith, Jason Statham, and Vin Diesel and that's about it. Even the Rock, who I thought might be able to lead an action revival has largely been relegated to family friendly fare. Today, you've got the multitasking actors like Matt Damon, Christian Bale, Daniel Craig who can do action, drama, and maybe comedy. Or you have an actor like Denzel Washington, George Clooney, Russell Crowe, Mark Wahlberg, Leonardo DiCaprio, or Brad Pitt who sometimes will do an action role.
As other posters have pointed out, women have also become action stars following Sigorney Weaver's lead. I doubt a film like Salt could've been made in 1980 or maybe even 1990, and be considered a credible action film. Plus we've had stuff like La Femme Nikita/Point of No Return, Alias, the Underworld films, the Tomb Raider films, Buffy the Vampire Slayer show, Terminator 2 and the Sarah Connor Chronicles, and recently Covert Affairs, among a host of other shows and movies that show females just as tough as any male, without all the macho stuff from the old action films. I put Angelina Jolie up there with the modern action stars of the 21st century, alongside any guy. She has a solid action resume, Salt, Tomb Raider, Mr. and Mrs. Smith, and Wanted. Milla Jovovich should be added on the list as well for Resident Evil, Ultraviolet, and the Messenger. Uma Thurman with Kill Bill. Jennifer Garner, Michelle Rodriguez, Kate Beckinsale, Rhona Mitra, Zoe Saldana, Cameron Diaz, and soon to be Maggie Q among others are continuing to change the face of the modern action film.
