How is Romeo and Juliet influenced by authority figures?

The play ROMEO AND JULIET is completely influenced by authority figures because the two lovers inhabit a world and era in which they have no power. We first see the clash of authorities when the Montagues and Capulets fight in the street, despite the Prince's ban on brawls. The Prince as ultimate authority decisively ends the quarreling between the two heads of families, but their young male relatives continue it. Romeo defies Capulet's authority to attend the party at which he meets Juliet and again when he trespasses into her family's estate to see her. The lovers as well as Friar Lawrence are defying all conventional authority (the parents, the Church, the Prince) to engage in a secret marriage. These acts of defiance set in motion the inevitable tragic end of the two lovers.

I am not sure whether you are asking how Romeo and Juliet, the characters, are influenced by authority figures, or how the play itself was influenced by authority figures in Shakespeare's world. I will assume the former since it is a more likely question. The main authority figure in the world of the play is the Prince. He states at the beginning of the play that the Montagues and Capulets will be punished if they fight again. He actually threatens them with execution:

If ever you disturb our streets again,Your lives shall pay the forfeit of the peace.

In the end, when Romeo does get into a fight, the Prince chooses to banish him rather than execute him. Romeo does not see this as mercy, however, since he has to leave Juliet behind. If Friar Lawrence had been able to complete his plan and send Juliet to Romeo in Mantua, though, the play could still have a relatively happy ending. It is Romeo's impulsiveness, rather than the Prince's word, that ultimately dooms the lovers.

It's also hard to know what would have happened to Romeo if he had stayed in Verona; indeed, he might actually have been killed by revengeful Capulets. So the Prince's cumulative effect on the characters' lives may be a neutral or even a positive one.