Virginia was the only member of the Otis family who noticed the forlorn and depressed ghost. Explain her character from this observation.

Virginia's interactions with the ghost show her to be a compassionate and pure person. If the rest of the family ignores, ridicules, and plays jokes on the ghost—or symbolically tries to rub him out through cleaning up his bloodstain—Virginia takes the time to really see and understand him as a suffering entity. Because of this, the ghost, already beside himself from having had to deal with her family, confides in her his sorry tale. He wants to die and have peace, but he needs someone pure like Virginia to intercede on his behalf and pray for him. She is willing to do this, and he is able, as a result, to go to his final rest.
Wilde chose her name carefully, and it too indicates that she is a pure (virginal) character. It's a double entendre, meaning it has a double meaning. It indicates both her Americanism—after all, Virginia is a state in the union—and her purity and innocence.