"She'd rather" in option A is incorrect as 'rather' is used to favour one choice over another. Here, no comparison takes place. So, rather is incorrect.
"She'd prefer" in option B is correct as 'prefer' is used to indicate choice without the need to compare with anything. In this case, there is no need to explain what other options can be used to reach home, such as riding a cycle, driving a car, going on the bus etc. But, 'walk' is the preferred mode over anything specified or unspecified. Thus, this is correct.
"She'd better" in option C is incorrect  as this is used to talk about actions which are desirable in a specific situation. Here, the speaker herself 'prefers' the action and desirability of the action is not under discussion.
"She has gots" in option D is incorrect as 'gots' is ungrammatical and incorrect.
Therefore, option B is the correct answer.