Which is correct:
I hope you and your family has a nice holiday season.
I hope you and your family have a nice holiday season?
In British English, "family" can be treated as plural because, in this sentence, you are thinking about the individual members of the family: "I hope your family have a nice holiday season."
This is common in British English: an institution composed of many people (e.g. government, team, etc.) is singular if it acts as a single unit ("The goverment has passed a law..."), but plural if the focus is on individual members ("The government have done nothing but argue").
In American English, the tendency is to use the singular, because you are talking about one family, one government, one team, etc.
However, in your example the subject is made up of two things: "you" and "your family". That makes it plural, so if you are a stickler for the rules, you should say "I hope you and your family have a nice holiday season".