If by “make a profit” you mean “be run like a private company with a profit paid out to shareholders” then I doubt that you can find a whole public trans. system, only individual things - bus companies like Greyhound etc., which can cherry-pick the profitable routes and times, and set their own fares.
If you mean “be run like a half-public company with certain rules by the city (to provide the service), but efficiently and break just about even, not with a huge deficit each year”, then several systems do this.
In my country, most public transport (local, not counting the train company, which desperatly wants to go and become a Stock Exchange company, to the detriment and anger of the passengers, but that’s another topic) systems are owned/ run by the city or quasi-city companies (that is, the company is partly or fully private, but the city has controlling majority or similarly).
That is because the public transport system is not there to make a profit (it should be efficient and not a waste, sure), but to provide a service to the people. And most certainly not for poor people - people here would be amused or offended if you suggested that - but for the whole population.
Everybody profits from less congestion and traffic jams in the streets (also less pollution) because instead of one single driver in his car getting to work, people take the subway or the bus.
Everybody profits from tired or drunk people using the bus at night after the pub or Oktoberfest, instead of driving drunk (taxis are expensive).
Most people see the sense in not driving a car into the inner city if you can’t get a parking space, or during the rush hour, unless you transport stuff, are invalid or live far in the outskirts.
Taking the local train for 30 mins. means time to read a book instead of getting frustrated with the rush hour traffic jam, so you have more quality of life and leisure time. 
This means that the city council tells the company to have buses running at 2 am when only two people use it, which is not economically useful, but has other values. The city tells the company to provide monthly and yearly tickets, which means that the busses are full but not a lot of profit is made. Of course, the company wants each bus full 100% all the time, but the passengers cry out and complain how full the busses are at 80% because they want to sit down and not be squeezed. 