I don't think there's any program out there that would take the sting out of fighting character art.
One thing to consider is exactly how many animation frames you need to do a particular move and what animation frames can be re-used. If you try and keep the frame count to a minimum, you could add motion blur to the fast movements. I would look at some fighting footage, a kung fu movie for example, and check out a move frame by frame. A lot of fighting games are designed to run at higher frame rates than a video so need more frames.
One major advantage that we have these days is alpha transparency. When SF2 was made there was no direct way to have a really nice motion blur, even with all the manga artists in the world, you can't motion blur without variable transparency. By using nice smooth alpha motion blurs you could easily get away with having much fewer frames compared to SF2.
Traditionally, I suggest you draw your sprites on paper then scan them in, maybe go as far as decent line drawings. Thing is that this sort of animation has to be tangible - movements have to be worked out, you have to have a good picture of the whole movement in your head. I think the only way to get that in 2D is to freehand the basic character structure then build detail from there. In 3D we have the benefits of seeing the thing from any angle at any stage in motion, so to do the same work in 2D you need a solid technique and a big chunk of talent.
One option would be to use a character model in 3D, and render the frames you need to 2D then draw over them. The character model could be a mannequin, like even one of those little wooden blokes artists have - something just to let you plan out the animation and render it to use as a guide.
It depends really on how you are at drawing and what style you go for. If you were more into Dragonball Z than SF2 then you might be better off with a vector art package, but to get the sort of detail an look they have in SF2 I think you need to rely on the old school methods.