Decent background = a family who has a good standing socially - ie parents, grandparents etc have a good repuation in the community...ie...honest and devoid of any scandals or accusations etc.
It would probably count as a positive if both parents live together (if alive ofcourse) and the family is in a decent position financially, not saying they should be Bill Gate's cuzins, but should be able to live a relatively comfortable lifestyle.
This is the definition of "decent background" according to my parents.