What is the likelihood that contemporary royals are influenced by public opinion?

Although contemporary royals are often more figureheads than the autocratic rulers they were in the past, the world is still obsessed with knowing the details of their lives. And when a young prince or princess is also beautiful on top of being wildly famous, that makes them even more fascinating.

Troll.

They are mindful of public opinion, as anyone would be in their situation, and employ media relations consultants to advise them. That certainly has an influence, but they take a broad consensus and are in it for the long game, so they tend to ride out adverse criticism. 

Camilla doesn’t care at all what the public thinks of her She didn’t care if people thought her a home-wrecker and a chain-smoker known to drink too much. 

They are. Witness the changes made after the death of Diana. Both Sophie and Catherine had a long courtship in which they were able to experience what it was like to be part of the royal family.

I think Diana's death demonstrated quite clearly that they are influenced by public opinion.  

Given that modern monarchies continue to exist, for the most part, solely by consent of the people, royals cannot afford to be inattentive to public and media relations. They have to keep an ear to the ground vis-a-vis the public's view of this or that issue, event, or behavior.