I am typing this from Uruapan in Western Mexico. MEX and pretty much the entire country (apart from Baja California) really are a lot better than their reputation seems to suggest. Crime of couse is a problem in Mexico, but it's organised crime, and I would say very unlikely to affect tourists (it's mostly narco-traffic and kidnappings of people known to be seriously rich- not just happening to be staying in a decent hotel). If you take normal precautions, you should be fine. I have felt very safe walking around the streets pretty much any time of the day/night- much safer than I'd feel in London, for example.

There are plenty of "pirate" taxis cruising the streets of DF, most of them are perfectly safe but I would still avoid them, especially if you're an obvious tourist- you're at a reasonable risk of being overcharged and a very minimal risk of something bad happening. Use the authorised taxis from/to the airport, and the hotel taxi and/or a reputable radio-taxi firm like Servitaxis for journeys within the city.

There are several authorised-taxi companies, with booths everywhere. I would say it's easier to find them than the guy holding a sign in a crowd of people!