As someone who's much better off as a person having a B.A. in Computer Science in five years than their intended B.S./B.F.A. in four, I think the both components are vital.

The undergraduate major I ended up with has definitely opened some doors; if I had finished the other half of my degree I wouldn't need to spend nearly as much time proving my literacy in other languages on the job. Most of my clients do business in English, so it's not a major disadvantage, but for someone who wasn't lucky enough to pick up a midwestern American accent at home, both the liberal arts side and the major can be vital for proving that you can do what you need to while you're not a "senior" or "highly experienced" candidate.