We been running back and forth to Houston for years and can't think of any particular stretch of road to caution you about. As anywhere in this country, big cities should always raise you alert level but otherwise there just isn't anything to worry about.
We even run route 9 between El Paso and Arizona right along the border with no concern - and that's a long lonely stretch of road. It passes through the historic little town of Columbus NM where Pancho Villa ordered 500 of his men to raid the town in 1916. If you are into history and have the time, it's a nice side trip about 30 miles south of Deming on I-10. There is a New Mexico state park campground there.