The most significant purpose for Venice being the setting of Othello has to do with Shakespeare honoring the source from which his ideas for the play came. Almost the entire play is based on an Italian author's writing.
The work of Geraldi Cinthio particularly focused on the struggles between husbands and wives. As soon as the play opens, the announcement of the eloping Othello and Desdemona takes the central attention of the characters. This is purposeful because the Venitians also routinely struggled through their marriages with loud jealousies and intricate quarrels. Not to mention, the overwhelming watch of a father would be a rather stereotypical Italian feature as well.
Since the play includes action that takes place in the ocean, it had to be a coastal city which likewise fits Venice. Othello is the military leader of the navy.
As in most coastal cities of the day, there was a variety of economic classes. This plays to the need for the play to have characters of great stature and political standing (Othello, Brabantio, and Desdemona) as well as the serving class (Emilia and Bianca). A setting like Venice was suited to provide this.
In the opening scene, Shakespeare portrays this variety of class as Iago eventually convinces Brabantio that he has been robbed of his daughter. Brabantio has pride for the high society that his city has and claims:
This is Venice;(115)
My house is not a grange.
Hope that helps.