I lived in Paris for about a year, and I can tell, if using common sense and not looking extremely tourist-alike there shouldn't be any problems.

But it's true pickpockets are a problem, there you have to be careful, especially when using public transportation. But IMHO nothing to worry about or feel unsafe.

But, there are areas in Paris, not really the centre, but towards the outskirts (something like the 19th or 21st arrondissement, I hope I did not mix the numbers as its quite some time ago I lived there), where mostly north-african immigrants live. I always avoided these place, especially by night, and this is what I would advise you, too.
Anyway, as a tourist, there is nothing, what could be of interest for you.

But don't worry, places that you will / should visit are generally very safe. Make yourself some nice time in Paris and enjoy your trip, I really love Paris!!!!!!