I'm also worried about this but the thing is, I think most people, regardless of background, are alright.
There are videos on YouTube and articles about it but most of them only list a couple of occasions where anything overt has happened. That's a very small number compared with the amount of time you will spend there and the amount of people you will meet.
Overall, 37% of students at Durham are from private schools. However most of these will still be fairly ordinary and will not have gone to the top private schools in the country like Eton. Many of them will be from families that struggled financially to send them to private school or could only just afford it. Durham is not going to be full completely of the "ultra rich".
My sister is at uni and has had problems with rich friends who invited her to go to expensive restaurants that she couldn't afford. However this is a minor issue and it's not like rich kids are going to go to restaurants like that all the time. If you wanted, you could only be friends with other state school pupils anyway.