When the actress Mia Katigbak was a student at Barnard College in the 1970s, she was the only Asian-American studying theater and mostly got to play “maids and hookers,” she said. One day the department head asked her to join a Molière comedy. She would play the harpsichord from behind a screen, Ms. Katigbak recalls his saying, “because there were no Asians in France at that time.”

No one is putting her behind a screen anymore. Today Ms. Katigbak, known for tenacity and grace as a stage actress, is much in demand. And for the last 25 years, as co-founder and artistic producing director of the National Asian American Theater Company, she has made it her mission to send Asian-American actors wherever the theatrical canon may lead, from “Our Town” to “Othello” to “Antigone.” Molière’s plays, too.

Starting July 6, members of her company will populate a fictional Bronx tenement when their acclaimed 2013 production of Clifford Odets’s 1935 play “Awake and Sing!” begins performances at the Public Theater. A classic of American realism, it follows a year in the lives of three generations of a Jewish family in cramped quarters. Asian-American actors — Chinese-American, Japanese-American, Filipino-American, Sri Lankan-American, Indian-American — will play the family, the Bergers, with Ms. Katigbak reprising her role as the abrasive, guilt-tripping matriarch, Bessie.

As Bessie tells her rebellious son, “If I didn’t worry about the family, who would?”

Offstage, Ms. Katigbak worries about family, too — the actors and playwrights and directors who are her artistic kin. On a recent afternoon, she sat in her company’s tiny office, its walls decorated with photographs of productions and a decayed bird, a prop dating from a 2009 production of “Seagull.” She spoke softly and precisely. She seemed a little tired, and no wonder.