There's no amount of convincing that is going to make them feel safe, you need to get them to use it, and change their opinion that way.
They're not gonna use it unless you build it, and they're not gonna use it unless it becomes more convenient than the car.
Right now you've got 3 unreliable bus lines, each bus 30 minutes apart, service stops at 8pm and the schedule is useless because they get stuck in traffic. Consequence is, nobody uses it and there's always a crackhead in the back (they're part of the population that actually uses it to get around even when it's inconvenient)
My point is, you can clean busses regularly (and you should but -), you can put a cop on every bus, you can do a lot of things to improve "the feeling of safety", but it's not going to offset inconvenience, and you won't need to do all that if you just make public transport the most convenient way to get around in your city. Except cleaning them. You'll still need to clean the bus.
Yes, this is absolutely true! But if you make transit convenient without addressing the crackhead in the back, the first time someone tries it will probably be their last and we're back to "nobody uses it, so we're cutting the schedule to save money".
Fundamentally it's a competition between public transit and the alternatives (usually cars). Cities can influence both ends - make driving less pleasant by cutting road infrastructure, parking, add fees for driving into the city, etc and make transit more pleasant by improving schedules, cleaning, etc. I'm a big fan of the latter instead of the former because the former often pisses voters off and leads to a backlash that sets progress back instead. Unfortunately, the latter often costs more as well.