Depends on where you're working, etc., but in general, West Side is where you want to live.
My mother lived in the Kern Place neighborhood near UTEP and medical district... nice older neighborhood, kind of in transition now so don't know if it's still a good place to live.
She lives a bit further out on Mesa now near Coronado High School ... decent neighborhood, neighbors are pretty much a**holes but that's the way it works.
I grew up out in the Upper Valley -- back then it was kind of separate from the rest of El Paso so was like its own little town, but it's all grown together now. If I were moving to El Paso, I'd look in Upper Valley first.