Time for me to repeat the old curmudgeonly rant about how we're infantalizing kids these days:
I started riding the NEC by myself when I was 10. My parents dropped me off at 30th St. and my grandparents picked me up in Baltimore. On one trip, I was dropped off at Baltimore by my parents, but I had to transfer upstairs at 30th St. to the Paoli Local to go to a family friend's house, where my parents, who drove up after me, picked me up. I also had my first meal in a dining car on that trip -- ordered and paid for it myself, with no adults to help me. I mean, learning how to spend US money was something I had figured out by 3rd grade.
At age 12, my parents put me on the Pennsylvania-Reading Seashore line to ride to Atlantic City to spend time with my grandparents. By the time I was in high school, I was joyriding on NEC trains to Trenton or Wilmington, and on the PRR main line to Paoli. My parents were OK with this, but said If I wanted to go to New York, I would need to go with a friend, which I did a couple of times, but I also disobeyed and rode up myself a couple of times for the hell of it.
I would get off in Penn Station (which seemed like a gleaming, glamorous terminal of a science-fictiony future back in, say 1970), ride the subway up to Times Square, go upstairs, look around at the flashing lights and billboards, go back down, ride back to Penn Station, and ride home. Keep in mind, this was circa 1970, when urban crime rates were as high as they've ever been, and Times Square as a sleazy mess, although possibly of interest to an adolescent boy, certain not a suitable place for him. But I never had any problems with crime or sleazy people or whatnot. Of course, my regular commute to school every day was on the Broad St. Subway in Philadelphia, which went through the worst slums in town, and I guess I had learned techniques for dealing with potential trouble, like, don't make eye contact, look like you know where you're going, etc.
Of course, the PRR/Penn Central NEC trains had a much better class of clientele on average than the subways, plus there were conductors always passing through. I never heard of personal security being an issue on those lines.
I think it would be an excellent experience for your daughter to travel alone to learn how to take care of herself, and an Amtrak day trip is as safe as anything can be with that. (And I'm speaking as the father of a daughter who rode Amtrak alone to and from college.)