It's important for a society to have a healthy institution of the press, as that is one of the channels through which government finds out about what's going on in society. Because both the government and the people are reading (etc.) the same content, it is common knowledge, which is good: the people know what the government knows, etc. (Compare this with domestic surveillance by the government, or Bentham's Panopticon.)
Therefore, because it benefits the public, the press ought to be partially supported by the government (because government acts in the best interests of the people). Can this be done without the government influencing the press? Freedom of the press is also important.
Some possibilities, low-hanging fruit. These aren't all necessarily good ideas.
Government support of various infrastructure needed to do journalism: the internet, public education (writing, reading). Open records regulations. Regulations requiring private organizations not to conceal information. Tax deductions for donations to journalism organizations. Trademarks so that trustable journalism organizations can establish a reputation. Copyright so that journalism organizations can earn revenue independent of government support.
This is made even more complicated because there may be some natural forces encouraging monopolies in the press, and monopolies are not healthy: economies of scale, tendency of power (to influence people and the government) to become consolidated, feedback effect of popular news sources using their popularity to become more popular.