The Other Subjunctive
Why is “It seems to be fair that he keep the money” better English (even if archaic) than “It seems to be fair that he keeps the money” (even if accepted modern usage)?
Jonathan Hammond responded to this question on Grammar Girl’s Facebook page:
It’s all to do with grammatical mood. In this case the mood is subjunctive. In the US it’s common to use just the bare infinitive but, as you have noted, in Britain we tend to add in a ‘should’.
Both versions are considered standard. The subjunctive mood, which uses the plain form “keep” instead of “keeps” for the third person singular, is considered the more formal version.
Why use the formal version? As Jonathan mentioned, it is more common to express the concept of mood in Modern English with the use of a modal auxiliary verb such as “should”. Mood expressed this way is called “modality”. Mood expresses concepts such as obligation, necessity, permission, ability, and degrees of remoteness from factuality. Unlike other languages, we hardly express mood with word endings anymore—it has become a lost art of sorts.
But by using “fair that he keep” rather than “fair that he should keep”, you are demonstrating to the discerning reader that you appreciate one of the more subtle distinctions in English and possess a robust knowledge of language in general.
Note that for anything other than the third person singular (I, you, we, they), we would use “keep” anyway, and there is no distinction between a plain declaration such as “They keep trying” and the subjunctive “It is important that they keep trying”. Not only is the distinction subtle in Modern English, it is rare and becoming rarer.