How many people are there in your family? This is correct and would sound fairly natural. Less formal: How big is your family? You could ask the second way and it would also be correct. You just wouldn't hear it as often. You might hear it asked with a particular inflection if the person asking was surprised at how big a family it was: "How many people do you HAVE in your family?

Thank you,JMat. I love your answers.

Danyel, 'there is/are' is used to indicate that something exists. As Frank says, it doesn't make sense if you interpret it literally. It's similar to 'es gibt' in German.

I meant the family where father, mother, and children

The "there" in the first sentence is unnecessary. How many people are in your family? Now the problem becomes a cultural one. Do you mean "current nuclear family", "nuclear family of your childhood", "extended family", ...?