I am traveling with my daughters, granddaughter (eight months), and son in law in Paris. My daughters have traveled in Paris two and three times before this time, I only once. I understand the safety code of being aware of surroundings… practical in any city. I guess I am just thinking about traveling with a stroller and baby in tow. Also, I am a bit concerned about taking the metro everywhere. We are staying Left Bank near Vaneau metro stop…. we will each have a bag of clothes - no valuables that seem vulnerable. I am feeling worried, should I be?
It won't much matter that you look like distracted tourists if you actually are organized and keep your valuables secure. Deep-store most of your cash and cards in a moneybelt under your clothes, keeping out only the cash or single credit card you need for a day or half day on the town.
Rick's Paris book includes a short chapter on ideas for traveling with children. Among them: the bus can be easier with a light stroller than the metro (which can have a lot of stairs). Taxis are also pretty affordable within the city, including those that are minivan-sized.
The metro is a bad idea with a stroller -- it means carrying the baby and stroller up and down countless stairs. On the other hand, the buses kneel, have a place in the middle for strollers and are easy to use.
There are really no safety issues in Paris. What there are is pickpockets and so don't be leaving phones or Ipads sitting on cafe tables or keeping wallets in pockets or using backpacks for valuables or being inattentive with your purse. Someone who is flustered and fumbling around with stuff is an easy target for pickpockets, but with several adults and one stroller this shouldn't be an issue.
I would take a taxi from the airport although they taxi won't have a car seat. If you aren't traveling with one you might want to arrange with something like Shuttle inter for a car with a car seat for the trip in. With so many people it will not cost a lot more than the RER/metro and I would not do that with so many people and so much stuff.
During the day I would just be sure that valuables are secure and that you aren't carrying much money or more than one credit card and then not worry about it -- no one is going to grab the diaper bag or assault anyone. When you use the ATM have one person face out behind the person using it to discourage the gangs of petty thiefs that are part of organized crime in Paris. Groups of teen girls or even groups of kids are 'run' by Fagans to grab ATM money or pickpockets of the careless.
Paris is the safest city I have ever visited -- you can walk with confidence almost anywhere in the city at almost any hour -- just the petty thievery of pickpockets is a concern. There will be many people with babies and strollers.
We took our 11 month old to Paris when we were newbie travelers and had no problems at all. We actually got to skip the line at the Louvre because we had a stroller. They escorted us to the coolest elevator in the world. I don't remember taking the metro but I'm sure we did. We stayed in a central location and walked a lot. There are many families raising children in Paris. I actually think Paris is more kid friendly than many other large cities. Relax and have a great time!
Having raised a baby in Paris, I think Janet's advice is absolutely correct. By the way, your baby will be a nice conversation opener with both locals and visitors.