We left our passport in our room safe and kept a photo on the phone. I did take my US drivers license with me as "official ID" if that were needed.
I used a small wallet with only one credit card, Navajo easy card, enough cash for the day, and driver's license in my front pocket. The rest were in my room safe. I generally used tap-to-pay with my phone about 80% of the time so the wallet rarely came out.
Regarding safety - I realize I was VERY lucky, but this was my experience:
I lost my wallet in a small park across from Shakespeare Books. I didn't know it until I was far away looking for my Navigo Easy card. When we got back to the hotel room, I immediately canceled the card I thought was in my wallet (turns out to have been the card that was in the room safe, aargh!) and ordered a replacement driver's license for return. Then, I got a call from a guy who said he found my wallet and looked up my name on Facebook, saw what hotel I was in from my posts and called. We arranged to meet at Shakespeare & Co. He gave me everything back -nothing missing (but the credit card was now dead since I also canceled it!). Even the cash was all there. He refused a reward.
I have NOTHING but nice things to to say about the French, even Parisians!!
Regarding canceled credit cards, both Chase and Capital One immediately updated the cards in my Apple Wallet with the new numbers so I could still use them anytime Tap-To-Pay was accepted. Very nice!
Overall, we found Paris very safe - we did have a way of securing our zippers on our day bag so nobody could easily get into them when we were on the Métro or bus, and we just moved right past the petitioners without making any eye contact. Never an issue other than my own clumsiness in the park. Just be sensible and keep some back up in your room safe.