As far as snakes, I would absolutely love to see Rena, Gyalopion and Sonora, but I know my odds are slim. There are so many snakes in NM that would be lifers for me, I would be tickled to find just about anything. As far as arachnids, I would love to find some vinegaroons and tarantulas. I would be in seventh heaven if I were to find a gila, but would also like to see geckos and phrynosoma. I'm also quite partial to whiptails since I discovered a population of neotesselata here in Washington state a few years back, they have a special place in my heart
I left Cruces in 09 and much has changed. However, if still accessible, hit up the Doña Ana mountains just north of US 70 and east of I-25. The easiest access is via Jornada Rd, which leads to the Jornada Experimental Range. A wide variety of species can be found there and the area is relatively safe, sans possible dove hunters in the mesquite-creosote flats. Last time out there, the road is 80% dirt, though this could be changed.
If you hit up the local BLM office, you might be able to get a hold of Patrick Alexander. He was a PhD candidate under Dr. Donovan Bailey in the biology department. If you happen to tour the bio department, you can also try seeing if Dr. Bailey can put you into contact with Patrick. If Patrick is in the area he'd be a great source to possibly show you around should his time permit.
Other options are Baylor Canyon Trail, both the road and actual Trailhead. However, the area is now recently under the National Parks System if I recall. Alternatively, you can also visit Dripping Springs State Park. Both areas are in the Organ Mountains, west side. On the east side is Aguirre Spring Campground/National Recreational Area.Rena
might be tough to find around the Las Cruces area, but it can be done. My best luck with "farting snakes" has been off Highway 9 though towards the AZ border.
These areas should keep you from being bothered by BP and hunters, for the most part.