First, it should be stated that your assertion needs to be proved. Not all people would agree that social class is more important than race in determining a family's characteristics. I happen to agree with you, but in a world of competing claims, we should always seek to prove assertions or at least have good evidence.
With that said, there are several reasons why social class is more weighty in determining characteristics than race.
First and most importantly, we are living in a globalized world. Through modern technology, communications, and travel the world is a rather small place. So, wherever you go, there are tons of similarities. So, what divides people is not so much nations, borders, customs, or race, but wealth. The "haves" live a certain way, and the "have nots" live in another way.
Second, with social class comes a certain disposition and outlook. Wealth and education creates confidence and this frameworkshapes families more than race.
I agree with the previous answer in that this statement needs to be proved.
Social class and race are so intertwined that it is impossible to say that one is more important than the other to determine a family's characteristics.
In general, black families do not have much wealth due to their ancestory and try to live off of their income. They have difficulties accumulating any wealth and therefore have trouble with education-student loans, then have trouble finding a good job-experience/education, then have trouble with a good home. Basically, they have so much more trouble with social mobility.
White families tend to have familiar wealth and therefore can live off of it if they fall into a rough spot. Their families typically pay off their educational debts and sometimes even spring for a down payment on a house.