Perhaps the feeling of attachment itself is a characteristic that a child learns form his parents. Generally a child who is securely attched comes from a stable home where parents are healthy and secure. In the same way, a child who comes from an unstable home where the parents are insecure, hostile and/or unhealthy is likely to have anxious/resistant, anxious/avoidant or disorganized attachment to the parent. And because people of higher economic strata are likely to have the resources and skills to maintain a healthy household for the child, the results of the experiment reflected that. So it isn't that the child has a secure base but rather that the child is just entirely mimicking and learning from the actions of his parents.
This seems like a viable explanation for how children develop attachment patterns. I agree with the fact that if the parents are stable, it is very likely that the child will be as well. It might have to do with genetics and depending on how the parents dealt with attachment as infants, their children will behave similarly and develop similarly to an extent.