A fully comprehensive and systematic intellectual analysis of this question probably would need to consider one or more specific contexts, and issues such as the following in each context: What does "morally responsible mean" in a specified context? Is there any difference between "crimes" and "immoral actions"? Immoral actions toward other nations, or internally toward a nation's own citizens, or both? What kind of government, i.e., political system? Founded on what?
In general terms, Objectivism emphasizes: "A country's political system is based on its code of morality." (Galt's Speech, p. 204 in the Signet paperback edition of FNI.) This is as true in a dictatorship as it is in a free country. Even dictators need and crave a moral sanction from their victims. Objectivism refers to this as "the sanction of the victims," essential for sustaining a dictatorship and any other political system that violates individual rights. (A capitalist government, too, needs a moral sanction from its citizens.) FNI p. 181 explains: "Every dictator is a mystic, and every mystic is a potential dictator. A mystic craves obedience from men, not their [independent rational] agreement. He wants them to surrender their consciousness to his assertions, his edicts, his wishes, his whims -- as his consciousness is surrendered to theirs. He wants to deal with men by faith and force...."
FNI p. 179 explains: "Make no mistake about the character of mystics. To undercut your consciousness has always been their only purpose throughout the ages -- and power, the power to rule you by force has always been their only lust." FNI p. 180 mentions the issue of moral sanction: "But it cannot be done to you without your consent. If you permit it to be done, you deserve it."
One scene that I find particularly memorable in this regard in Atlas Shrugged appears in Part III, Chapter III, second third subsections. Dagny has just returned from a month-long absence following her plane crash in the Rocky Mountains, and she is informed that she is expected to appear on Bertram Scudder's radio program. She initially refuses, but then accepts after being informed of recent developments involving Hank Rearden. The short speech that she delivers on the radio program nicely concretizes her refusal to grant a moral sanction to the looters. The "intellectual cop" then cuts the broadcast off the air just as Scudder is knocking the microphone away. Upon returning to her apartment, she finds Hank Rearden waiting for her, and they talk further about moral sanctions. The scene ends with a dramatic description of how Hank came to learn where Dagny had been for the past month, whom she had met there, and what it means for Hank and Dagny's future relationship. As for the looters, neither Hank nor Dagny is quite ready yet to "shrug," but they're getting closer. Near the end of the scene, Dagny asks:
"Hank, could you give up Rearden Steel?"
Another memorable scene appears near the end of Part II Chapter VII, involving the sending of a passenger train with a coal burning engine into the long tunnel near Winston Station. The use of a coal burning engine in the tunnel was known to be prohibitively dangerous due to lack of ventilation and virtual certainty of suffocation of everyone aboard the train:
It is said that catastrophes are a matter of pure chance, and there were those who would have said that the passengers of the Comet were not guilty or responsible for the thing that happened to them.
This is followed by a brief list of passengers and the ideas they accepted and espoused -- mystic-altruist-collectivist ideas.
These passengers were awake; there was not a man aboard the train who did not share one or more of their ideas. As the train went into the tunnel, the flame of Wyatt's Torch was the last thing they saw on earth.
In a different context, the context of an outlaw nation at war with other nations, Ayn Rand expressed essentially the same basic ideas. Her answers are published in Ayn Rand Answers, pp. 94-96:
The issue of citizen responsibility also arises in the context of a free country. Our own U.S. Declaration of Independence expresses it as: "governments are instituted among men, deriving their just powers from the consent of the governed," and "that whenever any form of government becomes destructive of these ends [protecting man's rights], it is the right of the people to alter or abolish it."
Update: Choices in a Dictatorship
A comment mentions:
...innocent men, women and children born in slave pen dictatorships have two stark choices....
Dictatorships do, indeed, leave few choices for those residing in them -- and still fewer choices when the dictatorships initiate war against other nations. It is misleading, however, to describe the retaliatory use of physical force by a free nation against an outlaw nation as bombing "with impunity by enemies of the dictatorship." Strategically, the goal of a retaliating free nation isn't to target civilians per se, but to end the outlaw nation's aggression and its underlying causes. For more explanation, refer to a previous question on this website, here. It's also important to emphasize the context of retaliation by a free nation. Bombings by dictatorships are not sanctioned by Objectivism at all, even if the target is also a dictatorship. Dictatorships drop bombs for very different reasons and objectives than a retaliating free nation.