Will a man be held accountable for the sins of the woman under his responsibility?
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions and all those who follow his footsteps until the Last Day.
Every mukallaf in the basis of syarak regardless of whether a man or a woman who has reached puberty will be responsible for each of his/her acts, deeds, rewards and sins. They will not bear the sins committed by someone else. This is in accordance with the statement of Allah SWT:
وَلَا تَزِرُ وَازِرَةٌ وِزْرَ أُخْرَىٰ ۚ وَإِن تَدْعُ مُثْقَلَةٌ إِلَىٰ حِمْلِهَا لَا يُحْمَلْ مِنْهُ شَيْءٌ وَلَوْ كَانَ ذَا قُرْبَىٰ
“And no bearer of burdens will bear the burden of another. And if a heavily laden soul calls [another] to [carry some of] its load, nothing of it will be carried, even if he should be a close relative.” 
Allah SWT states:
وَلَا تَكْسِبُ كُلُّ نَفْسٍ إِلَّا عَلَيْهَا
“And every soul earns not [blame] except against itself,” 
Ibn Kathir when commenting on this verse said this verse tells the story of the Day of Judgement when judgement, punishment and justice are passed from Allah SWT. Furthermore, each soul will be rewarded based on his deeds. If his deeds are good, then he’ll receive rewards. And if his deeds are bad, then he’ll be punished. The reason is every soul (human) will not bear the burden (sin) of another. This proves the fairness of Allah SWT. 
We simplify this discussion into several bases as the following:
- For each person who has reached puberty, then his sins are his own, and not going to burden his mother, father, siblings or husband.
- Parents or guardians are considered guilty when they let their children sin without advising, preventing, or trying to stop the transgression, or even worse saying that it is right.
- Every family member should advise one another. This is the basis taught by the Prophet PBUH:
الدِّينُ النَّصِيحَةُ” قُلْنَا لِمَنْ؟ قَالَ: “للَّهِ, وَلِكِتَابِهِ, وَلِرَسُولِهِ, وَلأَئِمَّةِ المُسْلِمِينَ وَعَامَّتِهِمْ
“The deen (religion) is naseehah (advice, sincerity).” We said, “To whom?” He (ﷺ) said, “To Allah, His Book, His Messenger, and to the leaders of the Muslims and their common folk.” 
Ibn Hajar said: Advice for all Muslims means:
- Showing the path for them (Muslims) for their worldly and hereafter affairs needs,
- Helping them through words and actions, covering their aurah (weaknesses/mistakes),
- Fulfilling their desperate needs,
- Preventing harm from them,
- Giving goodness to them,
- Advise them to do good,
- Prevent them from transgressions appropriate with the time and place,
- Respect their elderly,
- Love the children, advising them with goodness,
- Avoid deception and jealousy towards them,
- Love something good for them for the things that he loves for himself,
- Hate for something bad to happen to them the same as he hates something bad happens to him,
- Defend their property and dignity,
- Call towards good character and conduct which is included as good advice,
- Follow the guidance of the Salafussoleh R.Anhum.
Some salaf strives and gives their best in advising that they sacrificed their worldly interests when they don’t even care.” 
Every believer must protect themselves and their family members from hellfire as stated by Allah SWT:
يَا أَيُّهَا الَّذِينَ آمَنُوا قُوا أَنفُسَكُمْ وَأَهْلِيكُمْ نَارًا
“O you who have believed, protect yourselves and your families from a Fire,” 
Syeikh Muhammad Sayyid Thanthawi in his commentary of this verse said it is intended for every person who confesses to being a believer to Allah SWT so that they will distance themselves from the hellfire by doing good and avoiding sins and their family by advising them, showing them the true road and asking them to do good and prevent them from committing munkar. Moreover, Imam al-Qurthubi cited the opinion of Qatadah and Mujahid which states the meaning of protecting ourselves is being mindful of our actions and protecting our family is advising them. 
 See Tafsir al-Quran al-Azim, 3/383-384
 See al-Fath al-Mubin 1/257
 See Tafsir al-Wasith, 476/14