Not the current Internet. The current "Internet" is too vulnerable. We need something more implicitly secure, and if desired, anonymous. Something that can't be filtered or censored.
Batman's gotham is also too vulnerable. Their police department is understaffed and underequiped (except for when the story demands) the general civilians are too afraid to raise their voices. The current internet actually sounds like Gotham but just that the joker is a government department.
You know Batman isn't the good guy, right? He's a fascist archetype that operates as a paramilitary force in service of and in reaction to a weakened liberal democratic state. He's about as jack-booted thug as you can get.