Most safety issues in the forest are a matter of weather and terrain. Lightning can be a hazard in the summer, avalanches in the winter, and hypothermia-inducing snowstorms practically year-round in the high country of the Pecos. The respective antidotes are early departures from exposed ridges; awareness of avalanche conditions; and good equipment, particularly when backpacking. Do not take cotton sleeping bags on backpacking or horsepacking trips in the high forest. A disproportionate number of search-and-rescue operations result from visitors underestimating the weather and using cotton sleeping bags that become useless when wet. Use something that will retain its loft. Be on guard for altitude sickness as well; much of the forest, particularly the Pecos, is high enough to cause altitude sickness in the visitor just up from sea level.
As already mentioned, petty crime can be a problem at some of the campgrounds. Incidents seem most common at the beginning and end of the school year, and also during the big-game hunting season, mainly in October. Sensible precautions help lock your car, don't leave valuables in plain sight, etc., but some of the break-ins seem purely random in nature and appear to be rooted in ethnic frictions. Some of these frictions extend into the small mountain towns around the forest; the tourist will definitely feel unwelcome in some of the small-town bars which is one reason why none are recommended here under "Drink". Keep a low profile.
fees/permits
Most of the campgrounds have fees for overnight stays; there may also be small $2/night fees for parking cars overnight at some of the more popular trailheads for backpackers, particularly around Cowles. Most other recreational use of the forest is free. Access to the Pecos and San Pedro Parks Wildernesses was controlled at one time by a permit system, but the permits have been discontinued and access is now free and unlimited. Seasonal closures due to fire hazard may occur in any and all of the national forest area, particularly in June and early July, and open campfires may be restricted during the spring. Inquire locally; the web site above generally does a good job of staying current on fire-related restrictions.