The correct option is
B stay
In order to fill in the blank with the most suitable form of the infinitive, we need to understand the meaning and usage of 'sooner than'. 'Sooner than' means the same as 'rather than', which also means 'instead of' or preferring someone or something over the other. If 'sooner than' is used in a passive (voice) sentence, we use the word 'to' along with the verb. In an active sentence, we do not use 'to' with the verb. The verbs without the marker 'to', are called bare infinitives.As the given sentence is in the active voice, we will use the bare infinitive here.
Option A - 'Stayed' is the past form of the verb 'stay'. As we need the bare infinitive (original form of the verb), we can't use 'stayed' here. Hence, option A is incorrect.
Option C - 'Stays' is the third person form of the verb 'stay'. As we need the verb in its original form, we can't use 'stays' here. Hence, option C is incorrect.
Option D - 'To stay' can't be used here as the verb in its original form with the marker 'to'. As the sentence is in the active voice, we can't use the infinitive with 'to' here. Hence, option D is incorrect.
Option B - 'Stay' is a verb in its original form and is a bare infinitive. Hence, option B is the correct answer.
'I would go home sooner than stay here.'