This idiom is used to express preference. If we use would rather, we must use the dictionary form of the action verb. If we use would rather that, we are introducing a new clause, that is, a new subject and verb, and we must use the past tense.
I would rather read than watch a movie.
(I prefer reading to watching.)
She would rather ask the dean.
(She prefers asking the dean.)
Ezra would rather that you smoked at home.
(Ezra prefers that you to smoke at home.)
I would rather that he didn't drive alone.
( I prefer that he not drive alone.)