MissyH - I live in Cleveland and take the Rapid often. It is generally safe. The cost is $2.25 one-way (with no transfers!) or $5 for an all-day pass. They also sell 5-ride (transfers allowed), weekly, and monthly unlimited passes.

The train is busy during rush hour (7:30AM - 9AM weekday mornings, 4:30PM - 6PM in the evenings), with mostly commuters riding the train during those hours. It also gets plenty of traffic before and after sporting events. If you ride it on a Sunday morning before a Browns game, expect people to be rowdy and likely drunk (but having lots of fun).

The train does go through some lower income neighborhoods, though I wouldn't say they are very dangerous. If you're a single female and want to ride outside of the busy periods, I would recommend sitting in the front train close to the operator. At least if anyone harasses you (which is still pretty unlikely), you'll be within sight of the operator. They also have police officers on the train many times. You should be just fine. Enjoy the ride (if you haven't already)!