What is the state government's job with maintaining hospitals?I learned it was one of their responsibilities but what exactly do they do?

Responsibilities vary from state to state, so expect that
different states will take more or less control in operations and regulation of
hospitals, often times based on how stable the funding situation is in that state.  That
being said, we often find public hospitals at the county level.  These are non-profit
teaching hospitals that take state funding to provide more health care access for the
population.

At the state level, there are certification
boards for the physicians themselves, and that audit services and finances at the county
hospitals.  In other words, one big responsibility of the states is oversight of the
hospital system to prevent waste and abuse, and to ensure an adequate level of
care. 

There are also hospitals more directly funded and
directed by some states, including mental hospitals and prison hospitals.  Most of these
receive their funding directly from the state (as many patients are unable to pay and
there is a public safety issue in some cases) and/or the people working there are
directly employed and paid by the state itself.