I can't think of a relevant biological context for that possibility,
not in practical terms in a human body, anyway.  Consider this,
however: if you don't have enough stomach acid, and "pathogens" thrive,
the biochemical activity will do some amount of harm in terms of
stealing electrons and rendering vital biomolecules dysfunctional.

No, in general they don't.
Think of it this way: damage done by oxidizing agents is usually
peroxidation, the formation of epoxides, and/or breaking of the molecule
chain, and/or chemical addition of different radicals leading to a stable
molecule.

Its easier to estimate probabilities when you look at the molecule and its
electron configuration.

Oxygen:

.O. one unpaired electron on either side

(thats why oxygen is usually found in the more stable form O2 - that is O=O
which means each of the two oxygens shares two unpaired electrons making 2
covalent bonds to 2 electrons each)

Hydrogen:

H. one unpaired electron

a clearer (though a little to simplified) display can be found here:
http://www.krysstal.com/images/chembond_atom3.jpg

OH- means " :O:H " that is 2 electrons shared between hydrogen and oxygen
and one extra electron pairing the single electron on the other side.. thus
the "-" charge.

In order to act as a quencher the molecule should be able to add one more
electron to its structure and still be a (somewhat) stable molecule, OH-
already has one added to it, so its not really reactive in that way.

Most antioxidants have a few conjugated doublebonds.. these can transport
the "radicalic" electron to the end of the chain, thus stabilizing the
molecule and making it less reactive.

Antioxidants in biological systems are usually seen in the light of slowing
or terminating radical chain reaction.
In so far the release of OH- ions has no effect in any way to my best
knowledge.
(Although I would not rule out that pH value can have some effect on radical
chain reactions or the reactivity of certain metabolites)