I've lived in San Francisco, New York, London, Hong Kong and Paris. I bought my first car at age 38 when I finally "settled down". I only used public transport in all those places.
The only time I was "stuck" using a cab was when I was really pressed for time or when I had been out too late. Both are very easy to avoid and let's face it, with a car seat-needing child, the latter is especially unlikely that you'll be out with your pals and forget to look at your watch. Most public transport systems run to midnight or 1am. I know where I am at that hour these days and it's not on public transport!!
I've also done public transport in San Fran, New York, London and Paris with my children.
While there is little risk of some sort of accident, security on the platforms and in the stations is key. Keep children away from the edges of the platform and keep your eyes on them, especially if you have an escape artist like I used to...
Pickpocketing is your biggest "threat". Simply keep your valuables tucked well away, not in your bag or in a back pocket. Being held up is scarier, if more unlikely. The best advice is to keep to where there are a lot of people. Don't ride in empty cars, specifically at night. Try to board with others. Wait for your train where there are people around.
As a single woman, as well as a mom, I have never had any problems. Just take those above precautions and you'll be fine. I'm not paranoid but it's automatic and don't think about it anymore.
Someone mentioned elevators. Depending on the city, that might be a bad idea. Isolated in an elevator could put you in a vulnerable position. Fold the stroller and use the stairs or escalators. Be sure to have a good quality baby carrier. if your children are potty-trained, or almost, be sure to pass by the restroom before setting out. For babies in diapers, perhaps do a change before leaving. The facilities in stations are usually less than ideal and may be dodgy and good luck finding a clean changing table.
Check about food and drink. What are the rules? In Singapore and San Francisco, eating, even a small child having a snack, can carry a heavy fine.
Just as a side note, often signals are cut when you go underground or in tunnels so your cell phone might not work, including no internet connection on your smart phone or tablet. Mentioning this to avoid counting on either to entertain children.
Another plus; children love public transport. We didn't move to the 'burbs but we bought a house at the perimeter so we too, use the car mostly too. Ever since they were really little, they like taking buses, trams and metros when we can. They like moving around, looking out the windows and checking out the crowd. Older children like to figure out where they are on the maps, and where they're going.
Kids wont associate transport always with being strapped down. Mine even ask to take the train instead!