When I play an E on the D string (2nd fret) my E string will vibrate so much it makes the E note sound awful. I am trying to get in the habit of muting it, but is this normal or could my bass be messed up?

When I play the D string open, the E string does not vibrate.

When the E string hear the E an octave above, it gets excited and vibrates by sympathy. It's normal. See the floating thumb sticky in the technique forum.

When the E string hear the E an octave above, it gets excited and vibrates by sympathy.

You make it sound like it's alive :P

But basically that's it. You can't escape sympathetic resonance. Fundamentals and harmonics will vibrate on other strings than the one you are playing, so it's a matter of properly muting the strings you don't play.

Thanks guys! I will definitely get my instructor to teach me a little about muting in our next lesson.