I know I am repeating what has already been said, but these are my 2 days in Paris. I am over 60 and travelled alone. I arrived at Gare du Nord on Eurostar and hope I looked Parisian with dark pants, dark shoes and a modest shirt. I had a zipped back pack and a large zipped bag for my luggage, and I strode out of the station by a side door where people were waiting for taxis, as if I knew where I was going. I didn't notice anyone hanging around and no-one interacted with me. A 15 minute walk took me to my hotel just south of Montmatre. I left my passport in the hotel safe, my drivers license is all that is needed to satisfy police if stopped. My wallet was in my right pants pocket and I wedged my smartphone in horizontally, so everything was tight and I could feel things against my leg. My coat with zipped pockets covered my pants pocket opening anyway. Later I walked towards Concorde and it was obvious that Parisians walk the streets of their city with confidence. No one looked afraid! Many women walking alone. I did see clipboard girls at Concorde but they did not appear to be aggresive.
Now, "the scam": Twice between there and Arc De Triomphe a dark skinned man "found" something on the ground. I did not slow down or even aknowledge him.
The following day I used the metro quite a lot and had no problems at all. I tried to stand so that no-one could come along my right side where my wallet was, and I also tried to have something at my back so no-one could open my backpack.
The only times I felt uneasy was walking between Moulin Rouge and Carrefoure in Pigalle after dark, to get back to Montmatre. I think Pigalle is a little seedy, what with all those sex shops, and twice I was spoken to by dark skinned men lurking on dark corners of Bvd De Clichy. I don't know what they were offering but again a polite "non merci" sufficed and I kept on walking.
I was also spoken to by one of a group of 4 men (again dark skinned) almost blocking the entrance to the small park below Sacre Coeur, just to the right of the funicular railway in lower Montmatre. They seem to be there day and night. Again I did not engage them in any way.
I think to sum up, remember what your parents must have told you when you were little and Don't Speak To Strangers! Try not to interact with anyone who is obviously not a tourist. Most Parisians have no reason to speak to you. Nothing much happens when you are on the move, it's when you are stationary that you need to be extra vigilent. Keep valuables out of sight. I put loose change in my left pocket and €20 or so in my shirt pocket, so I would not have to take out my wallet too often. Hang your camera around your neck and not over your shoulder. It's mainly common sense like when you are in any large city.
I really enjoyed taking in the atmosphere of Paris by walking the streets. You miss such a lot when using the metro. You must not be afraid of visiting Paris. Just be careful and watch for scams etc.