I’m currently serving the US Navy and would want to re-enlist onboard USS Midway. Do I have to set up some arrangement and how much will I pay for my family(spouse and parent) to get in?

I think you would have to contact the USS Midway directly but it's a museum so I'm unsure if they would do something like that.

I am sure they would not mind you doing your re-enlistment on board as long as you don't mind that there will be people walking around during it, because it is a working museum. I would definitely recommend contacting them to answer all of your questions.