It was several years ago when I visited, so rossmacd's experience may be more relevant. At the time, I was able to go up the steps and into the aircraft but not further. The flight deck was inaccessible (perspex screen) and the passenger cabin was blocked off.

I believe they do timed, guided tours which will let you have more access, but these are at additional cost to the Interpid museum itself, for which I would second the recommendation. Also has the prototype/development Space Shuttle Enterprise on deck (another extra cost)