Eiffel Tower, Louvre, Sacré-Cœur, and the metro lines that serve them are high-pickpocket zones.
Use a cross-body bag, zipped, in front. Never leave a phone in a back pocket.
Watch for the 'ring scam' (someone 'finds' a gold ring and offers it to you), 'petition scam' (clipboard distraction), and 'bracelet scam' (Montmartre). Walk away.