Between all the drama and world-saving that happens in Sailor Moon, it's easy to forget that Usagi has a family. While they make sparse appearances throughout the series, it isn't clear what happens to them beyond that. Usagi has a mother, Ikuko; a father, Kenji; and a little brother, Shingo. Whatever happened to them?
Ikuko is a homemaker, staying home to take care of the chores and children. She's a strict, but kind mother, who only wants the best for her kids. She scolds Usagi for low test scores, even locking her out of the house on occasion as punishment. Despite her low tolerance for Usagi's antics, she loves her daughter and understands that she's slowly maturing into a young woman. She also treats Chibi-Usa as one of her own after she comes to stay with the family, and later does the same for Chibi-Chibi, though she never finds out about Usagi's secret identity, Sailor Moon.
Usagi's father Kenji is a magazine editor and the breadwinner of the Tsukino family. He's just as kind as Ikuko, but never really scolds his children to the extent that she does. He does show concern when he learns that Usagi is dating Mamoru, but isn't overprotective of his daughter and allows the two to see each other. In the anime, he makes appearances in Season One and Season Two, but never reappears after that. Meanwhile, Usagi's brother Shingo is an elementary school student who picks on his older sister.
While he isn't shown much in the manga, he does make appearances throughout the original anime adaptation. He's has a fear of cats, which causes problems when Luna begins living with the family. While he does eventually come to like Luna and treat her better, the same can't be said about how he treats Usagi.
When Usagi ascends to her status as Neo-Queen Serenity, it's explained that the power of the Silver Crystal keeps her and everyone in Crystal Tokyo young and healthy. It's not clear whether this also applies to her family, as we don't know exactly when this particular benefit of the crystal began. We know Usagi becomes queen at 22 and doesn't age past that point, but we don't know when the aging process ceased for the citizens.
In Act 47 of the manga, Usagi gives an injured Artemis, Luna and Diana to her mother, and they come to realize they probably will never see each other again. This is the last time the readers see or hear of any member of the Tsukino family. In the following act, we do see the cats killed by Sailor Lethe, implying that Ikuko was killed to get to them.
Taking this into account, and the fact that Chibi-Usa hints she doesn't have grandparents nor an uncle, we can assume that none of the family lived to see Usagi take the throne. It's possible that they survived or were revived after Galaxia's attack on Earth, but this isn't likely unless they had strong Star Seeds.
Unless Naoko Takeuchi comes out and directly states what happens, the fates of the Tsukino family are up in the air. It's nice to think that they lived and became citizens of Crystal Tokyo, but without seeing them or someone stating this is true, we can't know for sure. It's a surprisingly grim outcome for such a happy family.
