When does money earned from a permissible profession become unlawful (haram) if the work requires presence at or supervision of impermissible acts such as listening to music, consuming unlawful food, and drawing animate beings, knowing that the governess does not participate in these acts but is compelled to stay with the children to monitor them?

If your work in raising the children of disbelievers is permissible and does not involve forbidden matters such as offering pork or playing music, then there is no blame upon you. When you encounter a reprehensible act, you must at least condemn it in your heart, given that your work requires your presence. What you have mentioned does not harm your earnings from this permissible work. It is permissible for a Muslim woman to work as aider (nanny) for the children of disbelievers if it is limited to permissible acts and does not involve degrading the Muslim. Children are not held accountable before reaching puberty. It is not permissible for a Muslim to offer pork to anyone, nor to play music. It is more befitting for a woman to remain in her home, and if she needs to work, she may engage in work that is compatible with her nature and femininity, while adhering to Islamic guidelines.