Well if its not ran by the government then the government should not give it money imo.
Then how do you fund public organizations that act independently from the government? No, I don't think you should categorically bar any instance of funding a non-government organization with government money. If it performs a good public service, and does so in an efficient, non-corrupt, and not-for-profit way, it's a good idea to fund it as it can perform its mission without the inefficiency of partisan gridlock. Sure there are other ways to fund such organizations, but they usually aren't enough, and these things tend to be worth it for the public good. If it's something people want, but it would suffer if corporate greed got involved, and would also suffer if the government could corrupt it or gum it up with partisan bickering, a non-government organization is the best option. We should be using them for a lot more.