I think he did it because he is loyal to Asami remember in the elevator when Kuroda and Takaba went to question the missing girl? He said something like he would do anything for Asami. :P
His main goal is to be a good loyal friend to asami and he knows that aki is important to asami some way he is being so scenarios of their relationship he asked him is the housekeeping Includes the nightly business and he followed him twice.
Kuroda very clearly didn't like Akihito when they first met and went as far as to express his opinion regarding Akihito to Asami. I think he comes to grudgingly like Akihito when they're looking for Mayu, so it's a combination of doing the right thing and knowing how Asami would feel if anything happened to Akihito.
He didn't like Takaba at first, but I think as time goes by, he's ...well...indifferent? He teases Takaba sometimes
C. It's the right thing to do.
Kuroda is in law enforcement, he is obligated to protect citizens.
Kurode doesn't like takaba so why did Kuroda protected takaba Akihito?( A) Did he do it because if he didnt, he has to answer to asami ? (B) he did it because he want something in return? (C) It's the right thing to do. (D) none of the above. Whatever answer you choose, please back it up with details.