How do cultures influence the shift from biologically-based sex to socially-constructed gender?
Cultures influence the shift from sex to gender because they determine what traits will be associated with each sex. This means that gender will be defined differently from culture to culture. Without cultures, there would be no such thing as gender.
Sex is, as you say, completely based on biology. The organs that a person are born with determine what sex they are. Sex, therefore, is not socially constructed.
Gender, by contrast, is socially constructed. Gender can be defined as the set of behaviors and attitudes that are expected of people who are of a given sex. These are not biologically determined. Instead, they are determined by our culture. Our cultures decide, for example, how aggressive men and women are expected to be. They decide how men and women are expected to behave sexually. These things can vary from culture to culture and can vary from time to time within a given culture. As an example of this, it is no longer seen as inappropriate in many parts of the US for a girl to participate in the sport of wrestling. Women participate even in such things as mixed martial arts.
Cultures influence this shift, then, by creating expectations that start to define what behaviors and attitudes are expected of men and women.