I think it depends on the anarchist you ask. Kropotkin's specter is still haunting anarchists, and his influence is, in my mind, a mixed bag. When I first started becoming involved in anarchist circles, Kropotkin’s name was said in the same way that Christians mention Jesus, or progressives in 2008 breathed the name Obama – before I ‘d read anything of his, I had gathered that he was the pinnacle of anarchist thought, his works were as yet unsurpassed, and we could all only aspire to be like him in some little way (I know, sometimes I really need to evaluate the company I keep!) Upon delving in to his work, I could see why so many people appreciated him, but as someone who had come to anarchism with already latent anti-civilization tendencies, I also saw his work as highly problematic.
The problematic part in broad strokes first: Kropotkin was a scientist who wrote his most renowned works between about 1890 and 1912, and his writing generally reflects the biases regarding concepts like science, technology, civilization progress and so forth. This is not something I share with Kropotkin, or many of his more modern ideological progeny. Kropotkin was a communist, albeit one who was interested in a decentralized communism, but in his communism, he shared with the Marxist-lenists of his day (and ours) an overwhelming obsession with the means of production, and liberating those means by the workers. He did not really ever call in to question the concept of production itself, or the idea of workers. His rose-tinted view of industrialism was fused with a longing for the pastoral agrarian life he was familiar with from having grown up as a privileged Russian aristocrat. Like many anarchists today tend to essentialize various oppressed groups, Kropotkin always held a somewhat distorted view of the pre-industrial farmer.
Anarcho-communists would likely site him as a more ongoing major influence on their thinking, as would folks with an affinity towards social ecology. In particular I tend to see Conquest of Bread, and Fields Factories and Workshops mentioned by folks of these tendencies. Since I am not of those tendencies, I will leave it to someone who is to elaborate on those affinities. What I will say about these works is that, as I said before, Kropotkin had a particular perspective on the topics he explores in those books – that technology and science would be tools of liberation, whereas I would argue that they tend to be the opposite – tools used to expand the reach of capitalism and the state in to our lives.
From my perspective, I think Kropotkin’s most important work is Mutual Aid: A Factor of Evolution. Though I appreciate the work critically, and I think that it is important for readers to contextualize the work in regards to the time and location. His perspective on progress is evident here in the very structure of the book, exploring mutual aid first in animals, before moving on to mutual aid in (the tellingly problematic) groups classed as “savages” and “barbarians,” mutual aid in the medieval period, and finally in his day. This clearly illustrates how Kropotkin thought of evolution and progress as somewhat linear, and it is not an atypical view for the period. What it does miss is that sometimes evolution means going backwards, when a mutation doesn’t work, it dies out ,and we’re back to square one. Additionally, some of his science shows its age, and he has a tendency to rely on the interpretations of non-civilized peoples by European explorers or anthropologists, in a time when anthropology was much more definitively connected to colonialist projects.
That criticism aside, he does effectively illustrate the pervasiveness of mutual aid, which is still a cornerstone of anarchist thought and theory. What he gets wrong in the details he gets right in the overall sentiment, despite his generally overly optimistic and utopian outlook. While Kropotkin saw technology and industry as being a part of this, it is also possible to use much of his work in defining an anti-civ perspective. Again, critical reading and understanding the context in which things were written is important.
One place where Kropotkin got it totally wrong was in his support of the Allies during World War I, most notable in his written work in the “Manifesto of the Sixteen”. Kropotkin once again allowed personal biases (in this case the anti-German sentiment held by many Russians) to color his perspective, wherein he expected that an allied victory over Germany would further the anarchist cause. Clearly that didn’t work out so well.
Other works of his that are worth reading include his early writings agitating revolution, I have many of them in “The Selected Writings on Anarchism and Revolution” (MIT Press, 1970) and “Words of a Rebel” (Black Rose Books, 1992), and his autobiographical “Memoirs of a Revolutionist.” His early writings capture a different side of Kropotkin, one which was, at least theoretically, less gentle, more interested in instigating social upheaval. Memoirs is good in that it really captures his voice, and, I think, does a good job of illustrating how his politics came to be, and how they influenced his life.