A couple days ago, I went out to dinner with a group of Japanese law professors here in Tokyo. One of the Japanese academics, who today is generally libertarian, told me that he had previously been a Marxist. I asked him what led him to change his mind. To my surprise, he said that it was a result of reading Isaac Deutscher's books on Leon Trotsky. Deutscher was a Western Trotskyite who wrote a famous three-volume biography of Trotsky seeking to prove that Stalin had taken Soviet communism in the wrong direction, but that things would have gone much better if only Trotsky had won the power struggle between them in the late 1920s. The Japanese professor, however, deduced that Deutscher's critique of Stalin was applicable to communism more generally, not just the Stalinist variant. Thus, he quickly moved from Trotskyism towards giving up Marxism entirely.
Upon reflection, maybe I shouldn't have been surprised that Trotskyism played such a role in his transition away from Marxism. Trotskyism was also a way-station for many Western intellectuals who became dissatisfied with the Soviet Union in the 1930s and 40s, but wanted to cling to communism. Eventually, many of them gave up communism entirely (Irving Kristol is a particularly famous American example). It would seem that at least some Asian intellectuals followed a similar path. In this very limited sense, Trotskyism had a positive impact on the world.
On balance, however, I still don't understand the fondness for Trotsky shared by many Western leftists (and even a few formerly leftist conservatives I have met). The truth about Trotsky is that he was a brutal mass murderer. Trotsky was responsible for the deaths of hundreds of thousands of innocent people during the era of War Communism (1918-22 [corection: 1918-21]). Together with Lenin, he (not Stalin) established the Gulag system, the secret police, and other major institutions of Soviet repression. Trotsky also played a leading role in engineering the first, abortive collectivization of Soviet agriculture - which led to a deliberately engineered famine that killed several million people in 1920-21. Richard Pipes' book Russia Under the Bolshevik Regime has a good discussion of Trotsky's role in these and other early Soviet atrocities.
As bad as Stalin was, it's possible that Russia and world would have been even worse off had Trotsky defeated him in the late 1920s. After all, Trotsky broke with Stalin in the 1920s in large part because he thought Stalin wasn't going far enough in repressing "bourgeois elements," collectivizing agriculture (which eventually led to an even bigger deliberately engineered famine in the early 1930s), and promoting communist revolution abroad. In exile in the 1930s, Trotsky argued that the Soviet Union should not ally with the western democracies against the Nazis because both were "capitalist" powers, and neither was preferable to the other. Had Trotsky won, life would have been better than under Stalin for members of the Communist Party; Trotsky was less interested in purging the party comrades. But it might have been even worse for everyone else.
Western admirers of Trotsky often praise him for his criticism of Stalin's purges of the 1930s. However, as Leszek Kolakowski points out in the chapter on Trotsky in his comprehensive history of Marxism, Trotsky had no objection to political repression as such. He was very much in favor of ruthless persecution of non-communists, including even non-communist socialists. Trotsky merely objected to the repression of his own followers. Praising Trotsky for opposing Stalin's purges is a bit like praising the Ku Klux Klan as champions of free speech because they oppose laws banning racist hate speech. Obviously, The Klan would have no objection to censorship if they could be the censors themselves. The same point applies to Trotsky - except that he murdered, repressed, and censored far more people than the KKK ever did.