If you are visiting a military historic site, don't allow your kids to treat it like a playground.
Don't treat any historic site as a playground.
I do think, however, that behaviour should be determined by the age of the site. For example:
- 9/11 memorial: some of the people there have lost spouses, parents or children to the disaster or they themselves may have gone through a terribly traumatic episode. Act as you would in a graveyard: low voices, absolutely no running around, young children kept at arm's length distance to their parents (or, as grandma would put it, a ear's pull away).
- WWI memorial: it is at least one generation removed from us, so a little more freedom. I would expect people to talk normally, but still be very respectful. For example, it is not the place to tell a joke about life in the trenches.
- Civil War battleground: similar to other historical sites. I would not blink at a guide telling the more humourous side of army life of the period, but I would also expect them to explain the significance of the site and the sacrifices made.