Interesting question.
I approach this question from a multitude of angles.
I'm going to age myself here, but, as a child I was very angry when my Mom wouldn't let me watch "Psycho" (the original Anthony Perkins version). I remember when it was "ON" TV (some time in the sixties........back then you had to watch things when they were 'on', or else you missed it.), and I was sent to bed, while my older siblings were allowed to watch.
I was really angry!!
When I got to finally watch it years later, I couldn't understand what they were previously trying to protect me from.
But as for Indiana Jones, I don't think there is any age that you would need to 'protect' any child from watching the films. There is nothing in any Indiana Jones movie that would cause any kind of emotional or psychological damage to any child of any age.
And really this opens the discussion of what are we all trying to protect our children from anyway.
I know of people who won't let their teenaged children watch "The Simpsons"!! Seriously!!
You would think that people would be more concerned about the healthcare of our children than any kind of silly notions about Hollywood movies negatively impacting them.
In fact, you can take children who have experienced horrific things in their life.............children in the Balkan wars, ........child soldiers in Africa.......children who survived Hiroshima and the bombings of Dresden.......and despite their terrible emotional scars, they can carry on, and become healthy adults.
So I think that maybe our fears of what our children watch on TV or movies is a bit ridiculous when compared with the terrible things that so many children in the world suffer.
Though he is NOT a "Childrens'" character, Indiana Jones is nonetheless fine for children to watch.