I simply wouldn't camp. Coming in a RV, I would book a decent hotel room in the Miracle Mile area, store the RV near one of the eastern commuter stations (or nearby eastern stop on Amtrak, like Niles or South Bend) and take the train.
From the closest places you can stay in your camper, not only will you lose a lot of time commuting daily, the schedules will mean that you must miss the city night life. That might be OK for D.C. where you are going in to visit public monuments, but cities like Chicago, NYC, San Francisco the night life is a big part of why most people visit, and it helps to stay in the middle of it. London, Paris, Tokyo as well, but you wouldn't be tempted to try visiting by RV, most likely.
It is not cheap, maybe $200-300 a night in Chicago, $300+ in NYC, to be where you need to be for theater and shopping, etc.
Actually, I visit Chicago periodically to get my big city fix, and I don't go by RV at all. Air tickets are a whole lot cheaper than the cost of moving my RV 1600 miles, round trip.
As for big city safety, having lived in Chicago and Beijing, walked the night streets of London, Paris, Rome, Prague, Budapest, it is partly about not getting mixed up in criminal activity, and partly being aware of where you are, and when. For size, many of our mid-size cities (Tulsa comes to mind in my case) have higher rates of robbery and murder, but it still comes down to not messing with the drug dealers, and staying out of gang fights. Even then, half of what happens is family trouble.