Woman Jogger in Paris

I think it's totally safe. Lots of joggers at the Luxembourg Gardens if you care to go that far.

Your biggest hazard will be traffic. Most of the shady people selling trinckets on the Champs de Mars arrive by mid to late morning. Go a couple of blocks in the opposite direction, and you may find some more open space around Les Invalides.

Paris is not a great city for running, but it's doable.

There's quite a nice green promenade running along the south side of the Seine from the Pont D'Invalides to past the Eiffel Tower. There's a nice wide path or 2 that runs through it....hard-packed earth base with small crushed gravel...almost like an old-style cinder track. It's usually pretty open during weekdays.
I usually find the trinket-sellers fade out after you get past the immediate area of the Eiffel Tower....they're non-existent by the time you reach the Ecole Militaire....as is most pedestrian traffic on weekdays.

Susan - I've jogged around Paris and particularly in the Eiffel Tower, Champs de Mars area. It's very safe. I'd recommend going fairly early in the morning when there will be other joggers out, people walking dogs and sitting on the park benches, and before the sellers and tourists arrive.

I will never forget seeing the obviously French woman jogging in her running clothes and perfectly tied scarf!