To reasonable people, this seems like an easy problem to solve:
"We represent an esteemed science publication. If you wish us to publish a paper, we will also publish all of the data which led to your findings via easily accessible download. If that is a problem for you, perhaps you want to consider a job in a less demanding field than the sciences. Perhaps politics."
And then carry through.
The essential grounding of scientific thought and philosophy is that experiments must be replicable. If an experiment cannot be replicated in physical nature, the data which led to those findings must be able to be analyzed by someone else to arrive at the same conclusions. If you, as a scientist, are unwilling or unable to provide that data to someone else – you were not engaged in science. You are engaged in making stuff up and writing it down. If you, as a publisher, are unwilling or unable to demand the data which leads to the conclusions in the papers which you publish and tacitly endorse as science, you are not, in fact, publishing science. You are letting people make some stuff up, write it down, and then hand it to you to publish with less oversight and due diligence than a fan blog about Pokémon.
It really is that simple. The problem is a general unwillingness to hold bad scientists and bad publications accountable for the crap they produce. Part of that is the political environment of academia, where actually achieving or discovering new things that has been devalued in favor of simply producing more papers. Part of that is the social environment of academia, where the idea of actually challenging someone who might be wrong is seen as a terrible breach of social boundaries.
Until that environment which is socially toxic is cleaned up with whatever flamethrower equivalent you can find, this is going to be the norm – not the exception.