Any way to batch stabilize video files?
I think the quick answer is no, not that there is not software to do it, but every clip is different and will need to be stabilized using different paramaters.

If you take the easy way you may realize that after the fact it took way longer in that it could be a step backwards rather than actually helping.

I would recommend editing the clips as is, then stabilize when you are finished, that way it is only stabilizing what is used which is a) faster, and b) more efficient as you may cut around large camera moves that could cause un needed blown up footage.  At this point you could do each clip using software like premiere, which has a 30 day free trial.  And at worst you would just need to apply it individually to each clip, and unless this is a feature, it cant take that much time.