Did the people around Polonius tend to give weight to his opinions? Were these people significant?

Two significant people, Claudius and Gertrude, come to Polonius for advice and give weight to his opinions. After all, Polonius is the consummate courtier, which means he is a man who knows how to appeal to people in power, solve their problems and do what he needs to do to get ahead in the world. Since we come into the play in the middle of the action, after Hamlet's father has been killed and Claudius...

Two significant people, Claudius and Gertrude, come to Polonius for advice and give weight to his opinions. After all, Polonius is the consummate courtier, which means he is a man who knows how to appeal to people in power, solve their problems and do what he needs to do to get ahead in the world. Since we come into the play in the middle of the action, after Hamlet's father has been killed and Claudius established on the throne, we don't know how or why Polonius rose to the status of king's counsellor. But we know he is an important figure at the court,  and such status would lead people to take his opinions seriously.

Claudius and Gertrude seek out Polonius after Hamlet's erratic behavior begins to worry them, so they clearly value his opinions. After all, a son with possible mental problems, especially when he is a prince, presents a sensitive situation, one they would want handled quietly. They would approach someone they trusted. When they do come to him, Polonius presents himself  to them as the man with the answers, insisting that Hamlet is acting oddly because he is love with his daughter Ophelia. Polonius concocts a plan to spy on Ophelia and Hamlet when they are alone together as way to test his theory, and the king and queen approve his scheme. These two listen to him because he has established himself a go-to person for solving problems.

Polonius and Claudius share similarities, the similarities of wily, experienced players who know how to survive in a corrupt court. For example, they both hire spies to watch their respective sons. (In Claudius' case, of course, Hamlet is his stepson.) It seems no wonder that Claudius would value the opinions of a man who reminded him of himself and conducted business in similar ways. 

I'll add that when Polonius comes to king and queen with his news about Ophelia and Hamlet, he asks Claudius his opinion of him (ie, of himself, Polonius) and Claudius replies, "As of a man faithful and honorable." (Act II, ii) I don't want to ignore that Polonius is a pompous windbag, but do want to emphasize that important people listen to his advice, even if it's wrong. He is very much an insider.