Once upon a time, fear about crime on public transit was focused on subway systems. Although I cannot speak for every city in America, that certainly isn't true today in New York, Washington DC, San Francisco, or Los Angeles. In all those cities, underground mass transit has a reputation for being quite safe, whereas it is quite common, on asking a resident why they don't ride the bus, that safety is mentioned as one reason.
This squares with my anecdotal experience. Riding the DC subway everyday to work for roughly a year, I never once witnessed a crime or even felt threatened by another rider, whether on the subway car or on the platform. In contrast, my very limited experience on DC buses -- roughly two dozen trips during my two years in the city -- included witnessing a robbery, a small fight, rowdiness on two occasions that rose to the level of making me uncomfortable (I am not particularly sensitive either), and several instances at bus stops in marginal neighborhoods where I felt far less safe than underground on a subway platform (regardless of its neighborhood).
Although I've had a terrible time finding data about bus crime in Southern California, where I live now, the one relevant paper I did locate indicates that Los Angeles residents once cited safety as the number one reason for not using the bus, albeit at a time when LA crime rates were much higher than they are today.
Ridership among women is particularly depressed by the perception of crime.
I've been surprised by the dearth of data I've been able to find about crime on buses, so if any readers are aware of a relevant study, please do pass along the link. I'd also be curious to hear about your perceptions w/r/t bus safety.