When you say "the health problem" I assume that you are talking about the fact that so many people today are unhealthy (especially, if we are talking about the US, obese). If so, I would say that the family contributes to this because it is within the family that most decisions about what to eat are made and most of our habits are formed.
If you grow up in a family where your parents are busy all the time, you may well grow up eating unhealthy foods. You may get a lot of fast food and a lot of processed foods. Both of these tend to help cause obesity. As another example, if you grow up in a family that eats lots of red meat and lots of fats, you are likely to continue to teat those and are, therefore, likely to have problems with cholesterol.
Because our families make many of the decisions as to what we eat, they impact our health tremendously.