Why does Shakespeare choose Venice for the opening scenes of Othello?
Ha! Great question!
To call a woman a Venetian was akin to calling a man a senator: both were considered prostitutes of sorts.
In Shakespeare's time, Venice was considered the pleasure capital of Europe, and Venetians were known for their sexual licentiousness. The most talented courtesans were reputed to have lived in Venice. What is more, Venetian courtesans did not dress as we would picture prostitutes dressing. Instead, they were virtually indistinguishable from the average well (expensively) dressed woman.
Here is where it gets interesting. Desdemona is a virtuous woman (she finds it unthinkable that women would consider cheating on their husbands) and yet she is a Venetian by background. During the time period, Venetian women were stereotypically considered likely to cheat. Iago alludes to this repeatedly, and uses it to his advantage.
Part of the reason that this play works so well is that we are so busy looking at the racism directed towards Othello, we are almost blind-sided by the prejudice against Desdemona. The saddest part is that Othello is as blind as all the rest.
Arden Shakespeare. A.J. Honingmann ed. 2007
I agree with both the other answers, that Shakespears chose Venice, it seems to be a synecdoche for vice. But also, Shakespeare often set his plays outside of England, in order to guise the social commentary inherent within them. To make a comment on the social system of England, would be seen as unpatriotic, so he disguised his critiques of Elizabethan and Jacobian societies, by setting his plays in other countries. Many of them are set in Italy, or like Macbeth, nearby countries such as Scotland (granted though, that could have been a result of the original nationality of Elizabeth I's successor).
Shakespeare depicts a thriving commercial society in which the inhabitants pursue luxury and see the world in mercantile terms, as for example, when Iago says that he knows his own "price" in the play's first scene.