I think empathy is only bad if it leads someone to be counterproductive towards achieving a best outcome (I know, "best outcome" is vague and can vary based on context). From a personal-interaction perspective, I think empathy is generally a good thing. Relating with another person's perspective has benefits. Most of these are related to psychological and social well-being, and it leads to people not wanting to do harm to others. I can't imagine living in a world where no cared about how others feel. Imagine how depression and crime would skyrocket. Anyway, having empathy doesn't mean you're required to enable...it just makes it a little more difficult to make purely rational decisions. I think we should strive to have empathy while being rational about how to best help people. Even having empathy when you can't help someone provides perspective on others' experiences. Just because empathy can make someone feel guilty and/or powerless doesn't mean we shouldn't have it. In my opinion, only someone in a privileged position can say we should tone down our empathy. It's like saying, let me enjoy my privileges and not have to feel guilty about it.
Maybe I'm not thinking about it correctly, but I don't really understand why there are no "good" or "bad" emotions. As the OP said, they're just neurological states. But they evolved to be there for a reason...evolution made us prefer experiencing some emotions over others. For example, most people favor things that make them happy and hope to avoid things that make them sad. Perhaps on a philosophical level, they are just two states of emotions that can lead to positive, neutral, or negative outcomes (i.e. seeking happiness can lead to overeating, drug use, dangerous thrill-seeking, etc.). But if we seek one and avoid the other, aren't we "wired" to perceive one as good and the other as bad?