Normally I go rescue the settlement hostage (and Fast Travel back to the quest-giver ---- I escorted the hostage all the way back ONCE, but never again).
Most recently, though, I did not feel like bothering with it, and I am sitting on tens of thousands of caps since I usually sell more than I buy, so I just went ahead and PAID the ransom. Or, gave the farmer the money to pay it, I suppose. The mission completed as normal, but the farmer said something that made me think:
"I hope the raiders stick to their word".... or something to that effect.
I know the hostage-taking thing is just a random Radiant style quest that is generated every so often, but that line made me wonder if there were any implications to just paying the ransom instead of actually performing the rescue myself. Probably reading more into a random quest than Beth would have thought to program... I know I'd feel bad, though, if I later got a mission stating that the raiders took my money and KEPT the hostage, and the guy delivering the money, or something like that.