It definitely doesn't hurt. But I've seen awesome animators who are self taught with no art background. There's always an exception to everything, of course. If you have the time and opportunity to get a background in art, then I would go for it. For personal reasons and enrichment I would at least focus on figure drawing and character design-- basic areas that will help you create appealing characters and poses. If the school or institution is offering more than just fine art, I would branch out and go beyond the traditional art background.
Think about the art of animation as a whole and think about which aspects are important and helpful to your development as an animator. As an animator it's helpful to know at least the basics of acting, filmmaking, design, sound, etc. Basically most of the building blocks of a feature movie.
The study of motion and acting is the most important part to me. Good acting ideas and a good sense of timing will elevate you from being an okay animator to an awesome animator.
Filmmaking and design are important in order to understand the process of how to structure a movie, how the flow of sequences work, the composition of shots, the staging of actors, and so on. It's not just about cool animation. It’s about how you direct the audience's eye so that they are looking at the right character and not some tree in the background, how motion works between two or more shots so that cuts are not jarring and confusing to the audience.
You also have to think about sound because at the end of day the sound designers and mixers will add all the audio goodies to your shot. So if you have the main actor in a live-action movie deliver a line and you're animating a creature (like his pet dragon for instance) standing behind the actor, then don't have it roar or do crazy stuff, because that sound would overlap with the actor's dialogue and distract the audience and give the sound mixers a headache. :)
Think about the movie, or tv show, or game or whatever the project you're working on as a whole and not just animation.