Drifter I would say it depends. I have seen news stories that you will be charged with trespassing if you enter a national park that is closed. I can't tell if that means the ones which charge a fee or not. Seems strange since we already own the park to be charged for tress-passing on ones own land but that is where we are at in the modern world. The park no longer belongs to the people it belongs to them and they can shut us out.
In reality though if you can park outside the park and walk in then you can probably fish all you want and won't get in trouble. I would be big $ that if they saw you they will simply ask you to leave and not charge you with tress-pass. The reason being charging the wrong person with that (someone willing to fight it in court) there is a good chance they might lose in court then in future shutdowns it would be wide open for anyone to walk in and do what they please.