Several methods I can think of. First, just leave the printing stage off, let any libraries that find that important just print off their own copies, and remove that excuse for high fees. That's like version 0.01.
After that, you can experiment (which is sort of being done) with proper reputation systems to replace the "we're a big organization with $X, no one else can play" model. Sure - the big organizations would still dominate most of those, and scoring 'points' in such a system would still require money - but that money should hopefully go more towards people doing work, and less towards organization fees, licenses, etc. That would get you to something like version 0.35.
Getting to this point would involve lots of scandals - but proper ones that really should happen. To get further, you'd want replace the flawed "because we're older and got more mentions" system with a proper interactive vetting process, where replications are worth a larger percentage, even if they don't get 'published'. You can start to bring the newer system into the hiring process instead of 'must have published in x or y' process we've got now. That would get you around 0.5.
To get further than that, you'd have to get outside parties interacting with the process better. Imagine a world where not only free access, but journalists would actually use it, because it's mostly as convenient as 'industry sources' info. That, and being able to contact often obscure scientists to ask a question without having to wait for days in administrative limbo as often.
I'm not coming at this from a 'oh, why won't they support my pet topic' perspective - but as someone whose had friends that have had to deal with the system as it has existed, and who is into proper James Randi-style skepticism (not "science skepticism"), who sees flaws in journals and journalists covering topics lead to mass public misunderstandings greater than just a few simple scandals.
Any system is going to have flaws - I just don't see the journals as useful to anything at this point, when expert gatekeeping can be done so much better in other circles.