I don't know that I agree that the world did nothing and the movement withered on its own. I think the world governments did something and they just did not recognize what they were doing.
The anarchists appealed to social and political injustice as the things they wished to make right. And the their cause resonated with many people who were adversely affected by the very real injustices. World governments at that time were in the process of alleviating those injustices already. I think it was probably that alleviation of suffering and injustice (not elimination of course - but a definite reduction) that led to the cause drying up. Starving people facing unjust imprisonment are easily recruited - reducing the number of such people reduced the anarchist recruiting base.
Of course, one exception to the western governments who were doing that was Russia. And the bolshevik-anarchists carried their cause to its logical conclusion there. Governments that addressed the injustice escaped that fate. The US came closer than what many think as evidenced by the strong showing of Eugene Debs in the 1920 presidential race.
Interestingly enough, this is the sort of approach that John Poole recommends for drying up the recruiting grounds of Islamic terrorism in northern Africa in his recent book.
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