It is possible to talk about the "family business" in a generic way:
I grew up working in the family business
However, "business" by itself is often too ambiguous to tell the reader anything interesting about your family. Instead, it's natural to be specific about the kind of business your family does. Some examples:
Our family business is textiles.
Our family has a large share of the textile market.
Our family is big in textiles.
You can substitute whatever details are appropriate:
I grew up in a family whose business was making shoes / selling cars / international finance / technology / shipping and trade / etc.
Side note: I use "was" to talk about the family business rather than "is". This should not be interpreted to mean that the family is now "out of business". It simply refers to the past time period "when I was growing up". The family business could be doing just fine.
If you want to be clear that the family is still in the same business, you can instead use "is"
I grew up in a family whose business is making shoes
[Edit] The expression "business class" is not used outside of travel. If you mean to distinguish your family as something other than "working class", then the only distinction in common use is "upper class". Many upper class families own one or more businesses as their source of income, but it's their wealth that makes them upper class.
If you want to refer to the social class of people who own businesses (in something like an economic or political treatise) then there is the term bourgeoisie, borrowed from the French, which can specifically refer to the class of society which owns the means of production. It is not, however, normally perceived as a complimentary term.