While

doesn't have hydrogen ions in its structure, it's still a reducing agent:
Here Oxygen is reduced and Sodium Sulfite is oxidized to sodium sulfate.
Note also that there's always hydrolysis involved with any of these salts of weak acids:
and, of course
So in aqueous solution, there is always hydroxide and hydrogen ions present.
Oxidation-reduction
always involves something gaining electrons (reduction) and something losing electrons (oxidation), so it's not simply gaining a proton that causes the reduction. NAD may gain a hydrogen ion, but it also must gain one or more electrons to be reduced.