Paris is not much more dangerous than any other metropolis. The main danger for tourists here are pickpockets - they are active in public transport, the most touristy places (like the queues for the Eiffel Tower) and in shopping centers.
Do not carry large sums of money, wear a shoulder bag, do not shine expensive gadgets. Beware of scammers! They can offer you to buy something from hand, win a large sum in "thimble", "accidentally" get your clothes dirty and offer to clean them - there are a lot of tricks.