Mexico City is one of the world's great cities. More people should experience it.
Crime is mostly not a worry, the typical problem is the pickpocket in the subway. Avoid being in a scrum of people pushing into a car at the same time. This tends to be when this happens, and typically only at stations that are popular with tourists (Bellas Artes, Allende, Zocalo). The metro IS safe -- just keep an eye on your things. I've taken it all over the city.
The area you are staying in is fine. Lots of restaurants, cafes, people out and about.
I agree with the poster who suggested that altitude may be your biggest worry. It sometimes takes people by surprise who don't realize the city is at over 7,000 feet.