I got a ride from Sam Hughes (#575-436-2662) from El Paso to the official Crazy Cook terminus in '09. Joe Procella also offers shuttles in the bootheel area of NM (#575-654-6530.
Hitching is tough near the Mexican border - it took me three hours to get a ride on NM9 - but its always an option.
As Tony mentioned, the Columbus CDT route which finishes @ the border next to Palomas, Mexico is more convenient for public transportation. Corre Caminos Transit offers bus service between the border @ Palmas & Deming 3X daily. However, this is a busy, dusty border crossing - not as wild or scenic as the Crazy Cook terminus - which leaves you in the middle of nowhere…
Lordsburg & Deming both have Greyhound & Amtrak (‘Sunset Limited’ train) service.
El Paso, TX is the closest major airport, but I’ve found that Tucson, AZ (only slightly further away) has far more flights - especially on United Airlines.
Happy Trails!
freebird