The sentence indicates that the subject already knows Mr. Brown. Further, the sentence asks the subject, for what period of time does the subject know Mr. Brown. Thus, the answer must support the above mentioned points.
Option A is incorrect. 'do you know' inquires whether the subject knows Mr. Brown. Since it is already known that the subject does and the question being asked is for what period of time the subject known him, 'do you know' cannot be used.
Option B is incorrect. 'did you knew' is a grammatically incorrect statement. According to the rules of usage, if the auxiliary is in the past tense, the main verb cannot be in the past tense. Thus, it is incorrect.
Option C is correct. In 'have you known', the word 'known' indicates that the subject already knew Mr. Brown. Also, it supports the fact that the question is asking, for how much time has the subject known Mr. Brown. Thus, it meets all the required conditions.
Option D is incorrect. 'will you known' is a grammatically incorrect phrase. 'will you' is in simple future tense whereas 'known' is in the past participle form. A past participle can only be used with the perfect form of a tense. Thus, it is incorrect.