Considering the size and population, overall, I would say it's a relatively safe city. In nearly a decade I've 'only' had a handful of incidents where I've felt concerned for my safety. Which seems reasonable. Is that a lot? Hmm....
Women certainly do need to be much more mindful than their male counterparts in most countries. Even though Paris is very safe overall, there are a few things that female travellers need to pay particular attention to.
Whenever you're going to withdraw money from your bank account at home, make sure to use an ATM machine located INSIDE the bank. The machines located outside the bank are at risk of two things: 1. they could have been tampered with, and your data/funds could be at risk, or 2. someone could approach you when the bills come out and ask you for your money. The later has personally happened to me once in my early Parisian days. I was en route to London via the Eurostar. I foolishly went to withdraw money at an ATM outside the Gare du Nord and as soon as my bills came out of the machine a man came beside me and asked me for money. Thankfully I only withdrew a small amount, so I handed him a bill and walked away. I'm not saying this is the best way to deal with such a situation, but hey, I was young!
Cell Phones and Public Places
Make sure to keep your cell phone in your hands or tucked away in your bag when in public. Sometimes people leave their phones on the table at restaurants... BAD IDEA. There have been cases where someone will come up to talk to you, putting a book or magazine on top of your phone on the table, then, when they leave, they take your phone with them. Or they'll just run by and swipe it. Do not leave them on tables!
It's fairly common in North America to make eye contact with strangers as you pass by them on the street. It's seen as an assertive acknowledgment of some sort. In Paris this is not the case. Generally if you make eye contact with someone, it's seen as in invitation for a conversation. So unless you want to start chatting, keep your eyes averted!
The French generally don't smile when they're walking either and will give you looks if you do!
Scams at Tourist Sites
The main tourist sites are filled with scammers, and each area has their particular one. Some of the most common scams I've experienced are:
On the Champs Elysees: Someone will come up to you with a piece of gold jewelry, usually a ring, and ask if you lost/dropped it. If you take it or show interest they will ask for money.
Around the Louvre/Palais Royale Metro: People with clipboard will come to you asking if you speak English. If you stop, they'll try to get you to sign what appears to be a petition, but actually the last column asks for a donation. Once you've signed they'll expect you to pay up. Just ignore them and keep walking. They generally won't bother you unless you look like a tourist, so keep the fanny packs at home.
Around the Sacre Coeur: This happened to me a couple of years ago, so I'm not sure if it's still going on. Someone may come up to you and ask you to hold some strings. As you hold them they'll start braiding them while telling their life stories. At the end they'll tie it around your wrist, then tell you you owe them €20. Of course you can refuse to pay, but best to just keep walking.
Essentially if a stranger comes to talk to you IGNORE them. Unless they're asking for the time or for a cigarette from a safe distance.... even then, stopping to pull out your phone puts you at risk of theft, so to keep your visit safe, just shake your head "Non" and keep walking.
Pickpockets exist in most major cities. Don't tempt them! Make sure your bag has a zipper and keep it closed and tucked under your arm at all times. Cross-body bags are great too.
Entering your apartment building after dark
If you're renting an apartment during your stay, make sure to stay safe when entering the building after dark. Most buildings will have two doors. The first, outside, will have a code. MEMORISE THIS CODE. When you're entering the code at night, make sure to push the door closed behind you. These main doors close slowly and someone could slip in behind you. I've been followed home twice in the past decade while living in the 1st arrondissement. Better yet, if you feel you're being followed, try to go into a bar or hotel and ask for help.
Again, Paris is a very safe city overall, you just want to make sure you don't put yourself in any situations that could expose you to additional risk.