I've always liked the areas in and around Capitan. Beautiful area, somewhat remote, but has what you're looking for. Great organized match shooting group there too.
As stated, crime is everywhere. Some petty, some pretty damned violent. Has to do with the economic demographics of the population. Petty theft is rampant in my area near Dixon. Lock it or lose it. Don't know the exact statistics, but I had read that NM was among the top for highway deaths and drunk drivers. Heard that 50% of the homes in NM are mobile. Hell, I live in one too. I was down in Silver City for work and found that area to be pretty beautiful too. Mine economy causes swings in the real estate prices. They're shut down for now... I always liked Mule Creek and Mogollon. Reserve has some record Elk. Good luck on your search. Let us know where you land. H.