Yes, it is necessary for a woman to wear hijab.
And say to the faithful women to lower their gazes, and to guard their private parts, and not to display their beauty except what is apparent of it, and to extend their headcoverings to cover their bosoms, and not to display their beauty except to their husbands, or their fathers, or their husband's fathers, or their sons, or their husband's sons, or their brothers, or their brothers' sons, or their sisters' sons, or their womenfolk, or what their right hands rule (slaves), or the followers from the men who do not feel sexual desire, or the small children to whom the nakedness of women is not apparent, and not to strike their feet (on the ground) so as to make known what they hide of their adornments. And turn in repentance to Allah together, O you the faithful, in order that you are successful” (al-Noor 24:31)
O Prophet! Say to your wives and your daughters and the women of the faithful to draw their outergarments close around themselves; that is better that they will be recognized and not annoyed. And God is ever Forgiving, Gentle. (al-Ahzab 33:59)
As for what is the punishment, why should it matter what the punishment is? It is a muslim's duty to obey Allah, no matter what the command.
“It is not fitting for a Believer, man or woman, when a matter has been decided by Allah and His Messenger to have any option about their decision: if any one disobeys Allah and His Messenger, he is indeed on a clearly wrong Path.” (al-Ahzaab 33:36)
Do you suppose it pleases Allah subhannah wa ta'ala when we choose which of His commands to obey based on the punishment we will receive?
Please brother, encourage your sisters to wear hijab.