Both are correct. We use “had better” plus the infinitive without “to” to give advice. Although “had” is the past form of “have”, we use “had better” to give advice about the present or future. You'd better tell her everything. I'd better get back to work. We'd better meet early. The negative form is “had better not”. You'd better not say anything. I'd better not come. We'd better not miss the start of his presentation. We use “had better” to give advice about specific situations, not general ones. If you want to talk about general situations, you must use “should”. You should brush your teeth before you go to bed. I shouldn't listen to negative people. He should dress more appropriately for the office.

Both answers are correct. The more proper sentence would be "You had better do what your doctor tells you to do", but it is acceptable to leave off the "to do" at the end.