towen84! Hello! First, I'm going to recommend you go to the best website ever re Rome: roninrome.com It has more pictures, more info, more advice than you can imagine.
I've been to Rome about 10 times. I usually stay in the Vatican neighborhood (Prati) because I like it quiet at night after a long day of walking. Sometimes I travel with my girlfriends (all in our 60's) and we feel very safe there, and love many of the restaurants which mostly have locals dining, as opposed to the "touristy" restaurants with higher prices and a lower quality food. That's ME!
Once, because of an oversold flight, we ended up going back to Rome for another night (this time with the husband) and stayed near Termini because of convenience to get back to the airport the next day. Neither one of us liked it! Never felt like we were in any danger, but the locals were a bit "shabbier" than in other parts of the city, and unlike the previous poster, we had a heck of a hard time finding someplace we wanted to dine and enjoy our last meal in Italy! My husband likes the Trastevere neighborhood, and there's a younger vibe there. We also don't like to stay near Borghese -- it's mostly commercial there and it was hard to find a restaurant at night. DO take a look at Roninrome.com It's a fun and easy website.
So, different strokes for different folks! Ken and Frank are both very experienced -- but I don't enjoy this neighborhood like they do.