Is there a basis for the saying, "You are my sister by the covenant of Allah, and upon the betrayer be the curse of Allah," and what is the legitimate means for a man to be a brother to a woman in order to help her in matters of religion and worldly affairs?

It is not permissible for a woman to be a "sister" to you except through the brotherhood of Islam, for every Muslim is your brother in faith. Friendship between foreign sexes is not permissible in Islam because it leads to haram acts such as khalwa (private seclusion), looking, and speaking in a soft alluring tone, and everything that leads to temptation. It has been established that the limbs of a human being have their share of zina (fornication/adultery). The zina of the eyes is looking, the zina of the tongue is speaking, the zina of the ears is listening, the zina of the hand is touching, the zina of the foot is walking, and the heart desires and wishes.