Both are correct! "Family" belongs to a special group of nouns that can be considered either singular or plural.

"My family was in France last Summer"

i agree to you ホタル......"was" is the past tense form of the verb "is",while "were"is the past tense form of the verb "are".You can only use "is/was" if the subject is singular,and "are/were" if the subject is plural......

My family was in France last summer coz the word family in here is consider to be a singular so the subject linking verb should be singular also which is the "was".