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.