The first one, at first glance, seems quite strange— I'd rather (I would rather) you didn't come. — Better not come. What's interesting about this construction is that despite the use of past simple tense, the present is actually meant.
In the second construction with rather —I'd rather stay at home than go for a walk. In this sentence, the language learner hesitates whether after than an infinitive with 'to' or without it should be used. Please note that after than the infinitive is without 'to'.