How does The Master of Go present narratives in which gender strongly determines the way men and women act and are perceived by others?

Given that The Master of Go is a novel reverent of traditional Japanese culture, it is not unusual to find characterizations of gender that may seem out of place to a Western audience.
Japan has maintained a greater fidelity to the concept of patriarchy and the predominance of male authority in the home and family in comparison to European and American cultures, and The Master of Go reflects this specifically Japanese norm through its narrative focus; women do not figure as central characters in the story but merely serve as backdrops to the primarily male-centered action.
The story portrays the protagonists' wives as unendingly loyal to their husbands; the essential responsibilities of these women are reflective of the traditional role of women in the family as understood by the Japanese at the time the Go match was played.
For example, the Master's wife—both in chapter 7 and in the final chapter of the book—almost exclusively cares for her aging and infirm husband. She tends to his physical illnesses before he partakes in Go matches against Otake and schedules all medical appointments with the Master's doctor. Of course, this is not to say that such activities are easy, and the book even points out how the constant stress of caring for her husband's well-being puts a strain on her psychological health. In chapter 7, for example, the narrator explains:

The strain on the two wives [of the Master and the challenger, Otake] is painfully apparent to the onlooker. On August 10, for instance, during the play at Hakone, when the Master was desperately ill, the faces of the two women seemed drained of blood, their expressions were tense and drawn.

The consistent engagement with their husbands' performance keeps both of these women fully preoccupied and never short of issues or anxieties.
However, the narrative always relegates any individual virtue the wives may have to an insignificant level of importance in comparison with their husband's responsibilities. The Master's wife, for example, never actually watches his matches personally. Rather, she meets him as far as the game room in the mornings, patiently awaiting by the pond next to their room for the game's conclusion (which could take several hours) in the evenings.
These characteristics of the Master's wife conform to the traditional Japanese standards of family hierarchy: women are to be obedient, respectful, and submissive, dedicating themselves not to personal aggrandizement but rather the uninhibited preservation of their husband's honor.
In this way, the actions of both the Master and the challenger—including the manner in which they are dramatically portrayed during the Go match—are a normalized depiction of traditional Japanese values. That the overwhelming portion of The Master of Go celebrates the accomplishments and strategy of the two male competitors makes sense given the gender dynamics of twentieth-century (and traditional) Japanese society.