There are no major safety issues in this region. A few minor ones, however:
The north side of the range was the scene of pronounced ethnic conflict of complicated origin in the second half of the 20th century, and there are still residual Anglo/Hispanic tensions in some areas. Simply being respectful goes a long way to defuse these, but it's probably wise to avoid small-town bars on the north side, and to be alert at backcountry campsites there.
Please take restrictions on open campfires, etc., seriously. These mountains are flammable and have seen a number of nasty forest fires recently.
If backpacking or backcountry camping, purify stream and lake water, as Giardia parasites are present in water supplies, as usual. Tap water is OK. Be careful also not to get water in your nose or eyes if you're bathing at one of the "wild" hot springs; dangerously pathogenic amoebas have been found in some of the springs and can cause life-threatening brain abscesses. You probably won't have any problems from them, but keep your head above water just to be on the safe side.
In many regards the Jemez don't "feel" like high mountains, but they are, and the sun is intense; use sunscreen when outdoors.
Not so much a "safety" issue as a legal one: SR 4 west of the Valle Grande is notorious for radar traps and has numerous, basically inexplicable changes of speed limits that afford opportunities for traffic citations. Pay attention when driving here. DUI is a problem in much of northern New Mexico as well, and can be a concern in this region, although it's less of one than in the valley.
