कुलक्षये प्रणश्यन्ति कुलधर्माः सनातनाः। धर्मे नष्टे कुलं कृत्स्नमधर्मोऽभिभवत्युत ॥

kula-kṣhaye praṇaśhyanti kula-dharmāḥ sanātanāḥ dharme naṣhṭe kulaṁ kṛitsnam adharmo ’bhibhavaty uta

In the destruction of a family the immemorial family traditions perish; in the perishing of tradition, lawlessness overcomes the whole family;

In plain words: Arjuna says that when a family is destroyed, its old traditions are lost. When traditions are lost, the family stops following what is right and falls into disorder.