Some friends and I were actually debating this the other day XD My opinion: It has the potential to be a very close battle. (Also, spoilers for Serenity! Mind you, this thread in itself is kind of a spoiler for Serenity...)
For Buffy:
Buffy has superhuman strength, I'd say she could deal out more damage per blow than River, as I don't believe its ever stated that River has unnatural strength.
Although I wouldn't say her fighting style is as polished and natural as River's seems to be, after several years of using her fighting skills on a nightly basis, she has to have picked up a lot of tricks.
Slayers are hardy creatures, it would take a lot to take Buffy down and keep her there.
She has experience in battling a wide variety of enemies- demons, vampires, etc, whereas River's experience, we can safely assume (due to the nature of the Firefly universe) has only fought normal humans and Reavers (which I agree, being fuelled by rage and adrenaline, will be tough), though I suppose there is a possibility that she fought against other Alliance "subjects" for practice/testing/display before she was rescued by Simon. Thus, Buffy has experience with facing a lot of different styles. HOWEVER, it could be argued that, for all her hardiness and strength, she's still a human, with the same basic weak points of humans and Reavers, thus making this point somewhat more immaterial.
For River:
We know she's taken on a room of Reavers, killed them all, and still been standing by the end of it. We've seen Buffy almost killed by single vampires several times throughout the series run. Saying that, we see Buffy in vastly more fights than we do River, we'd really need to see much more of River before judging how easy it is to take her down (though going by rules of TV, I imagine that would vary depending on the needs of the episode).
She has an unnatural aim- in both her major battles in Serenity, and in the gun fight in War Stories, if River aims to hit something, with bullet, fists or knife, she generally does. I suspect this would include Buffy- and if River hit her enough times, or in the right places, or with the right weapon, then that could end the fight for her.
If we're talking River post-Serenity, her mental state appears to be a good deal more stable, and she no longer needs to be triggered by external forces, she uses her abilities of her own choice.
Lastly, and this is the big one: River is a psychic (or a reader, call it what you like). She can read people's thoughts, she can sense what they're about to do, and what's about to happen- see the beginning of Serenity for just one example. In a fight, this would be a huge boon- she could read what her opponent is about to do, where s/he is going to move, what weapons and tricks they have up their sleeve. I'm assuming this is part of the reason River fights so effortlessly, in many cases, not even having to look at her targets.
Overall:
I'm going to say River, the ability to read minds should theoretically win it for her, in my mind. Buffy can hit harder, and could take River down in fewer moves, so River would need to avoid Buffy's attacks- but if River can read minds, she should be able to predict most of Buffy's moves, and avoid them. She will need less recovery time, as she's proven she doesn't need to see her enemy to successfully land a blow, if she can sense where Buffy is, she won't need to be looking at her (for example, if she executes an attack that ends with her facing away from Buffy, she won't have to spend time turning and "locking on" to her target again). Buffy is very strong and fast, I imagine she would manage to do some damage of her own, or at least keep River on the defensive for a while, though. But yeah, my vote goes to River- assuming, however, that this is post-Serenity, and thus River has more control over her abilities... otherwise, if Buffy doesn't say "Miranda", I'd definitely say Buffy.
