Mexico, especially its big cities, has a crime problem. Be savvy about where and when crime could occur and keep yourself out of harm's way.
Be aware that bandits occasionally hold up buses and cars on intercity routes, especially at night. Deluxe and first-class buses use toll highways, which are less prone to hold-ups. Mexico City taxis are notorious for (sometimes violent) robberies. Don't take a cruising cab; phone a radio taxi service instead.
Pick-pocketing and bag-snatching are risks in large cities and crowded areas frequented by large numbers of tourists - wear valuables underneath your clothing.
Mugging is less common than bag-snatching but resistance may be met with violence, so give them what they want.
Leave most of your valuables in a sealed, signed envelope in your hotel's safe. Stash money about your body and only carry a small amount of ready money in a pocket.
Use ATMs only in secure locations and try to use them during working hours.
Don't accept food or drinks from strangers as there have been cases of drugging followed by robbery and assault.
Be wary of attempts at credit card fraud. One method is when the cashier swipes your card twice (once for the transaction and once for fraudulent purposes). Keep your card in sight at all times.
In beach resorts be aware of ocean riptides and undertows. Like in any country, lone women should exercise caution in desolate places, including beaches.