Why is reputation the most important thing to Brabantio?
Reputation is important to Brabantio for a couple of reasons. First he is wealthy and therefore a respected "aristocrat" in Venetian society. Second, he is a senator. So, Brabantio is a rich politician. It seems that even in Shakespeare's time rich politicians were obsessed with reputation and appearances.
Brabantio is "old-school" in the sense that he believes, because of his wealth and political position, that his daughter, Desdemona, is his property. She's young and beautiful and therefore a valuable commodity to Brabantio because he can marry her off to the "right" family.
Unfortunately, Othello and Desdemona mess up his plans by falling in love and getting married behind his back. To Brabantio, that's bad. His daughter has now defied his will/plans, gotten married without his blessing and permission to an older man who is black and perhaps not a Christian. Brabantio is upset at his daughter, but he's equally concerned with how it will reflect upon his reputation. He even goes so far as to accuse Othello of using spells or drugs on Desdemona to win her heart.