Suppose you had the choice of spending your life savings to save the life of the person you love versus spending it to save the lives of all those who died in the holocaust. Which would an objectivist choose? How is the choice to let hundreds of thousands of strangers gruesomely die at the cost of letting one person live who matters to you personally, morally justified?
This question understates the magnitude of the Holocaust and overstates what any one individual would ever be able to do about it and by what means. The Holocaust was not just some gruesome death of "hundreds of thousands" of victims by unspecified causes. Wikipedia describes it as follows in "The Holocaust":
The Holocaust ... was the mass murder or genocide of approximately six million Jews during World War II, a programme of systematic state-sponsored murder by Nazi Germany, led by Adolf Hitler and the Nazi Party, throughout the German Reich and German-occupied territories.
The Nazis were fueled by a fundamental, all-consuming totalitarian philosophy of racial supremacy -- mysticism, altruism, collectivism and statism in support of a presupposed "Aryan master race." Refer to Leonard Peikoff's first book, The Ominous Parallels, for further explanation.
It is ridiculous to think that the mere spending of money by an individual, even one's life savings, could do anything to stop a major, philosophically driven political philosophy like Nazism. The cause was philosophical, and ultimately only an opposing philosophy, implemented by America and its allies in a world war, could bring it to an end.
The question asks, in essence: suppose economic aid could prevent the Holocaust, somehow; what should more prosperous individuals do in that case? This basically turns the actual reality of the Holocaust into a fantasy of economic "power" by individuals, and asks for principles of ethics to be defined based on such hypothetical fantasy. In complete contrast, Objectivism does not derive principles of ethics from hypothetical fantasies (or emergency situations). Objectivist ethics is for normal real life. For the basis of Objectivism's life-based ethics, refer to The Ayn Rand Lexicon, topic of "Values" and related topics listed in the cross references.
There is absolutely no rational basis for an ethical principle demanding that one sacrifice oneself for others who have no personal or trade relationship with whomever is being urged to try to help, no matter how many allegedly "needy" others there may be nor how gravely they may be threatened. At most, rational individuals would want to work to remove the threat, particularly if it threatens them and/or their loved ones or trading partners. Removing the threat most likely would mean organized, governmentally driven military action as needed against the murders or torturers -- if the rational self-interest of the retaliating nation warrants such retaliation.
In any case, whether a rational nation chooses to retaliate or not, it is contemptible to hold them responsible in any way for the deaths of the murderers' victims. A rational nation is not the cause of the deaths; the murderers are the cause. A rational approach to cause and effect (and its ethical implications for man) never loses sight of what the actual causes are.
A rational approach also digs deeper to understand the underlying factors that allow the murderers to rise to power in the first place and remain there. It isn't an inexplicable accident. It happens because of the fundamental ideas (mysticism-altruism-collectivism-statism) that the people accept and follow, and their lack of better ideas. Physical force by tyrants only reinforces the deeper intellectual poverty of the victims. Otherwise, they would rise up in revolt on their own and overthrow the tyrants, or die trying.
Causality also applies as much to the creation of wealth in the first place as it does to death and destruction at the hands of murderers. A nation that prospers through rational initiative and productive effort is the cause of its own success. It has earned what it produces and is entitled to the enjoyment of its prosperity, regardless of how many other nations there may be in the world that do not strive to prosper through rational initiative and productive effort. That basic mode of survival -- rational initiative and productive effort -- is the essence of the right way to live, the only way that can bring prosperity and happiness in man's worldly existence. It is also the essence of what the victims of totalitarian governments most desperately need to learn. Merely removing the tyrants will not, by itself, make the people free to function rationally and productively. Without fundamental philosophical change, the underlying philosophical factors that produced the tyrants remain in force to do it again.