Apparently back in '02 the Japanese were considering a farm for the generally plentiful Minke whales. This strikes me as an excellent idea if it is practically feasible. By partially domesticating them and learning how to breed them for foodstock you could take pressure off of natural wildlife whale stocks. Research into their breeding was also a motivation for the farm, and that seems like it could potentially help in maintaining more endangered species as well.
So a refined question would be, if whales were sustainably farmed in similar (albeit underwater) circumstances as other domesticated animals like cows/chickens/pigs, would you eat that whale meat?
I imagine the majority of folks still opposed have a special view of whales, similar to how Americans view cats/dogs/horses and many Hindus view cattle. No harm in that I guess! I draw the moral I'm-not-touching-that line at fellow members of my own species, that's for sure.