There are no significant safety issues with the park itself usual warnings about wildlife, sunscreen, etc., but its remoteness means that you may want to pay a little extra attention to road safety while getting there and back. Northern New Mexico is notorious for problems with drunk drivers. Areas near Navajo Nation, as Chaco is, are particularly worrisome, as the prohibition of alcohol on the reservation causes residents to drive into Farmington or Gallup to indulge. Terrible accidents have happened involving Navajo Nation citizens on the way home after an evening of drinking; be extra cautious at such times.
Be alert also for livestock on the roads, particularly sheep. It's wise to fill your gas tank in Farmington or Gallup or Grants or Thoreau before heading for the park, as services are sparse indeed once you get off the main roads.
Weather conditions and hydration cannot be adequately emphasized. Every year there are rescues of visitors who are improperly preprared for either. From June to October, the Four Corners region is subject to violent afternoon and evening thunderstorms called "monsoons," or by the Navajo, the "male rains." The storms build quickly and can be preceded by strong winds, even into the 40-60 mile per hour range. Tents should always be tied with guy wires, and care must be taken ensuring the tent is not located in even the smallest of drainage courses.
It is safest to hike to isolated areas in the morning with the activity timed to return before mid-afternoon. When hiking during Monsoon season, carry a poncho or other rain gear. Take shelter and avoid trees or outcroppings because of lightning. If caught in a storm, if a feeling of hair raising or tingling is felt, crouch low to the ground, stay out of puddles, and try to keep your body from direct ground contact. Balance by holding on to your shoes. At your campsite, the tent is the safest shelter.
In terms of hydration, carry four to five liters of water when on one of the wilderness hikes. Anything less can be deadly. Drink before you feel thirsty. Set a rhythm where regular "swigs" of water are part of the hike. Even when visiting the "accessible" abandoned pueblos, it makes sense to carry at least a liter and to hydrate regularly. A good ratio is one liter of an electrolyte-filled drink to three liters of water. Consuming exclusively water can dilute eletrolytes and create a deadly body condition. For best results, use electrolytes that have little or no sugar or other sweeteners.
fees/permits
Entrance fees for the park are $4 for individuals, $8 for cars, and are good for seven days. The National Park Service's "Park Pass" $50/year secures unlimited repeat entries for this and other National Park Service units. Campsites see below under "Sleep" are $10/night, with a $5 discount for holders of a Park Pass.
Permits are required for backcountry hiking. They're free and available at the visitor center, or at the trailheads.