Scientifically speaking, there has been very little done on how much the abs and "core" do during running. I have a study going on right now where I am studying the muscle activation of some of these muscles during 5K intensity running, and will know in the coming month or two how these muscle fatigue during hard running. I can say already that they are certainly active during running (as they also may assist in breathing at high intensities).
I believe there has been one recent study which examined the effects of stability ball core training on running performance, which found no significant improvements... However, I am not familiar with all the results off the top of my head. I am also aware of an unpublished cycling study which found an abdominal training program did not improve cycling performance.
That said, these studies may be limited in certain aspects, such as what the training was, how performance was measured, the duration of the study, etc., etc., etc. One type of core stability training truly may be effective whereas another type may not be - there needs to be more science or maybe even mainstream anecdote before we really know.
Also, "core" training is very loosely defined. Some people consider crunches a core exercise, since it hits your rectus abdominus, whereas others do not consider one as it is not a "functional" movement. There are many different types of "core" exercises which are done in different positions which definitely seem more functional than crunches, and also hit other "core" muscles besides just a few abdominal ones. I think some of them make sense and can increase your stability while on the ground, and theoretically improve the way you transfer momentum throughout your body during running... but it hasn't been scientifically proven as far as I am aware, as of yet.
Hope that helps a little...