Yes, Bill De Blasio probably would have been a lot less successful without the support of his family. Being in the public eye can no doubt be stressful, but I'm sure that his family discussed all of the pros and cons of becoming public figures. His diverse family and viewpoint will help his entire community and family in the long run.
Family is the center of every person's life, whether intentional or not. Becoming mayor can be a very stressful and time consuming job but it is very honorable as well. He was elected for the people by the people and this was obviously something he wanted to do otherwise he wouldn't have run. It would be foolish if his family did not support him in accomplishing his personal goals.
I think that while there is a lot of stress and many doubts regarding a family member running for office, I do think Bill De Blasio's family was happy to see their husband/father/etc win the role of Mayor for the city of New York. They also all seemed really supportive of him.
Yes, I believe that Bill De Blasio's family was happy upon his election as mayor, because political connections are always a good thing. Somehow, high ranking officials somehow end up rich. De Blasio's family will win government contracts for their friends and other favors. Exemptions and attention because of a political family member is always good.
I do not believe Bill De Blasio's family would be unhappy as a result of his election as mayor, so I would have to assume they celebrated. His wife is noted as a poet and activist so his position actually helps her in her own work. The children, still attending school, I'm sure were also proud of their father.