A very good topic!
I think it had a LOT of influence as far as the trio is concerned.
We have no knowledge of Hermione's muggle background other than her parents were dentists. I think that her parents being doctors and being well-read influenced her quite a bit in that reading department. My idea of Hermione's house is place full of books and her parents putting pressure on her to keep up with extra-curricular reading. So her insecurity of being afraid of failure and always being on the top might have been born out of that. Add to that her exacting and critical nature.
As for Harry, I think he would have been a little more arrogant and less humble about him being the only one to survive Voldemort if he would have been raised in the Wizarding World. The unhappy and loveless upbringing with the Dursleys made him realize that love is more important than anything else. And that was preciesly the reason Dumbledore wanted him to be raised by the Dursleys!
It's been obvious in the books that most of Ron's insecurities are born out of him being in a large family where whatever he achieves is nothing extraordinary because one of his brothers has already done it before. Also, when he was destroying the Horcrux the point it makes that he was born when his mother wanted a daughter etc.
So, all in all, I think that their upbringing has A LOT to do with who they are although not exclusively. Of course who they are as a person also comes in to play.
Edited by redhotght - 28 December 2008 at 6:07pm