when to use these two sentences

Question

I would prefer that she do that

and

I would prefer that she did that

Why one is present subjunctive do and the other past simple did

Answer 1

I prefer she do that. [now]

Certain verbs take a bare infinitive (a better term than subjunctive). I recommend or advise she go now. Prefer, recommend and advise all take a bare infinitive in the third person.

I would prefer that she did that. [there is less chance of her doing it].

This is structured like: I would prefer if she did that.

I would like it if you did that. [an second or unreal [some say irrealis] conditional]. That is: If + simple past and the main clause using the present conditional.

Answer 2

Both are and can be used in everyday conversations but the correct one is:

I would prefer that she did that.

Cheers Travis