First, let me say that Paris is my favorite city in the entire world. I think it's safe if you use common sense and you are unlikely to get mugged or held up. However, it's been my experience and that of many friends and acquaintances that Paris has more than its share of skillful pickpockets who target foreigners. The subway (Metro) seems to be the worst place for it.
They often work in teams. As you get on an escalator going up, one gets in front of you, the other behind. When the escalator gets to the top, the one in front of you "accidentally" falls, causing you to pile on top of him and the one behind lifts your wallet in the confusion. Years ago, there were also bands of Gypsy children (directed by adults) who would target foreigners, but I have not heard of this recently.
I also had one guy approach me in the subway, unsolicited, ask to see my metro ticket, and give me the line "Oh, monsieur, this ticket is not valid. You need to follow me (pointing to some dark passageway) and we can straighten this out". I told him in French that there was no problem with my ticket and he got a panicked look on his face and walked off.
I would recommend securing your money in some fashion (money belt, pouch, etc. ) to keep it out of reach of sticky fingers. I always keep my wallet in my front pocket, secured with a large safety pin and just keep a small amount of spending money in my other pocket.
Just avoid these creeps in the subway and you will have a great time in one of the world's great cities.