Static characters are usually secondary characters. They are always sort of flat or one dimensional. They do not undergo any real sorts of changes in the course of the work of fiction. Don John really fits with this definition because he is just a bad guy. He does not really do anything in this play other than act like a complete jerk.
Don John does not show any real redeeming qualities in the play. He is nasty and antisocial. He tries hard to ruin his brother's life and he does a terrible thing (for no good reason) when he smear's Hero's name.
Since he spends the whole play being a bad guy, he is static.