My husband and I just spent 6 glorious days in Paris. Yes, there are pickpockets and other scams around, especially in touristy areas, and yet we had no problems, and I think this is why:
1. We both wore black shoes. Sounds dumb, but when we tried to pick out American tourists, we looked for white Nike-type shoes, baseball caps, and T-shirts with logos. No local people wear that stuff. We wore black shoes, neutral shirts (I wore colorful scarves) and black pants.
2. We both used money belts under our clothes. I heard sad stories about stolen passports/money/credit cards lost in a wallet or snatched from a purse. I carried only enough cash for the day in a small change purse in an inner pocket. Credit cards were in a money belt, along with my passport.
My purse had a locking zipper (which I used constantly), metal cables in the handle, and steel mesh in the body to prevent slashing. I think the scam artists recognize such purses because no one even approached me.
3. The 'gold ring scam' is alive and well. We followed 3 women along the Seine, each of them tossing a gold ring in her hand. As likely American tourists approached (white Nikes, baseball cap, and T-shirt), they would bend down and pretend to discover it. I was amazed at how many tourists tried to talk to them. Just say "Non, merci" and above all, keep walking.
4. There are very aggressive 'flower men' under the Eiffel Tower in the evening. They have carnations and will forcibly try to grab a woman's hand and force the flower into it. I jammed my hands into my pockets, said "No!" and kept walking. They didn't follow.
In short, although you can't blend in totally, you should choose your clothing with some care, always wear 1 or 2 money belts for cash/passport/credit cards, don't pull out your giant tourist map and look lost, and whenever approached just say "no" and keep walking.
Paris is a beautiful city. Use common sense and have a great trip!