New Mexico's residential permit guide says the building official issues permits to alter or improve a regulated building, and its list of work that needs no permit names things like painting and tiling, not basement insulation. That leaves the question open for your job, so I check with the office that covers your address before we start. If a permit is needed, we apply, set up the inspections through the city's Citizen Portal once it's issued, and keep you posted.
A basement sits below ground, and Doña Ana County is designated as having moderate radon levels, so it's worth testing before you close up walls or turn the space into a bedroom or office. New Mexico residents can get a free radon test kit from the state's Indoor Radon Outreach Program, one per household while supplies last.
We're based in Las Cruces and work across the cities around it, including Chaparral, Anthony and Canutillo, TX. Basement calls here usually start the same way they do everywhere we work, with a room someone wants to use more or a space nobody has looked at in years, and we handle them the same way.