Achenback Canyon: off Soledad Canyon Road outside of Las Cruces. Hike into upper valleys and goes up to a cave. Thank Brian and Bert for the suggestion.
Directions (thanks Brian):
Here are the directions from a Las Cruces trails book that I have.
Starting at University exit #1 on Interstate 25
0.0 miles Head northest on University (which eventually turns into Dripping Springs Road)
4.8 miles Turn right on Soledad Canyon Road and head south
.7 miles Turn left at fire station and head East (still on Soledad Canyon Rd)
3.5 miles Turn right on Ladera Canyon Road and head south
.6 miles Turn left on unmarked primitive road (next to dumpsters) and head East (road is a bit rough)
.2 miles Turn right into open area next to road and park