A fair concern Tony, and by extension I don't think that you would wish your children to adopt any sort of snobbery themselves, but as someone who went to a private school I think it's fair to say that those types are easily avoided at all but the poshest of boarding schools. The friends that I made are generally some of the most down to earth people I know, and are in the majority.
In answer to the thread it would be middle class. My parents are from pretty humble backgrounds but both managed to do well enough for themselves to afford me the comfortable lifestyle that I enjoyed growing up.
Class is not entirely dead though, and I would agree with Paddy that other than the absolute silver spoon dickheads, snobbery has almost become an issue of inverted significance.