An American friend who has lived in Paris for over 40 years, and who has seen her share of visitors come and go suggests these simple precautions:
Do not keep credit cards and cash in the same wallet.
The Paris Metro is relatively safe and you will frequently see rather small children taking it unaccompanied on their way to and from school and other activities. It is also frequently crowded at night, which is a good thing. However, if you are taking the metro alone at night and the line is rather empty, then choose to sit in the first car, just behind the back of the driver (excluding lines 1 and 14 which are driverless)
Major tourist areas are a magnet for pickpockets and organized gangs of thieves from central Europe. You needn't be frightened – Paris is no different from most other major cities – just be aware.