Try this:
Open Heart G Farms, Anthony, NM.2nd from the top. Looks to be quite easy off/on I-10. It's a horse motel w/RV E/W hookups and probably plenty secure. $20/night for the horse (indoor)... Doesn't say how much for a rig.
If you can hoof it (no pun intended), there are more horse motels/stables in Deming. If you're looking to save some coin, I'd go with the NM Rest Area. Most likely WELL patroled by local LEOs and/or State Troopers. Tell them you've been driving for 12 hours and really, REALLY need "a nap." Of course, you don't SAY that your nap might last about 8 hours, but they don't have to know that part. If you're decent with them, they'll usually cut a cowboy a break and just knock on your door and tell you to move rather than just tow you right outta there... With you IN the rig! I'd still say it's more secure than a truck stop lot... But some of their private security can be rather intense at times.
2ndary to this, RV2Go's Wal-Mart sounds like a winner, as well.
