It does. Kaioken x3 and x20 improve damage by roughly 15%
Also another interesting thing I found out while testing Kaioken:
Don't you just
hate how Kaioken time is wasted if your enemy manages to turn on the offense and whoop your ass until Kaioken time runs out and then you're utterly defenseless, causing even more damage?
Well worry no more,
You'll Never Beat An Android Z-soul's stamina recovery upon taking damage
works even with Kaioken x20 on! Getting hit by basic combos will let you regain stamina at roughly the same rate you're losing it, and getting hit by a basic Kamehameha can give you 3 stamina bars to continue your onslaught!