Well, minke whales are smart enough and social enough such that killing them is probably perceived in a very painful way by both the direct victim and their family. If you have enough empathy to think it's wrong to chop the leg off of a cat (which is vastly less social, less internally aware, and less empathic towards themselves and others) then there should be some empathy to these whales.
They're carnivorous, so their meat is slightly less healthy than it could be. But their diet is awfully low in the food chain, so is quite a bit more healthy than most of the meat we'd eat.
Resolution of this issue requires something that we don't have, and something that (weirdly, to me) a lot of people agitate against. International governance of common resources. The organisation that monitors whale hunting is largely toothless and terribly political. Minke whales could be more social than us, and we'd still have a hard time regulating their hunting.
I have a hard time seeing how whale hunting would be feasible (economically) without large subsidies. Maybe I'm wrong, but that's where I think there'd be broad consensus amongst all political stripes. Currently, fishing (and related) subsidies are obviously a giant problem. They cause overharvesting, they cause wasteful resource use, and they consume tax dollars that need to go elsewhere.
If we could get people to give two small craps about fishing subsidies, moral issues like we see here would be a lot less prevalent. But, as we know, partisans bicker about things that won't change, and won't actually form cohesion to stop things that they both despise.
