Prior to the collapse of the Russian monarchy and ascension of the Bolshevik party, it would appear that anarchists were treated as the primary scapegoat of Red Scares, and a good deal of conversation from "bourgeois" papers revolved around dissuading people from anarchism. They were targeted in the first Immigration Exclusion Acts, and subsequent Exclusion Acts as well.
Theodore Roosevelt seemed to hold a special contempt for anarchists:
“When compared with the suppression of anarchy every other question sinks into insignificance. The anarchist is the enemy of humanity, the enemy of all mankind, and his is a deeper degree of criminality than any other. No immigrant is allowed to come to our shores if he is an anarchist; and no paper published here or abroad should be permitted circulation in this country if it propagates anarchist opinions.”
-- (April 9, 1908)
The better distribution of property is desirable, but it is not to be brought about by the anarchic form of Socialism which would destroy all private capital and tend to destroy all private wealth. It represents not progress, but retrogression, to propose to destroy capital because the power of unrestrained capital is abused.
-- (September 3, 1910]
What I'd specifically like to know is:
Was there a widespread belief that anarchism and anarchists posed a significant threat to the capitalist order before the Russian Revolution?
This is the only real question I can talk about, so sorry. To answer it yes, anarchism is regarded as a major issue at the time, and many consider it retrospectively as the first fully fledged terrorist movement. (Terrorism: Critical Concepts in Political Science by David Rapoport is a pretty good read on this with the first volume entirely about the anarchist wave). Examples of how seriously people took this movement are apparent both from quotes from leaders at the time (look at how harshly Roosevelt talks about anarchism) as well as reactions to events like the Haymarket Affair, where anarchists were sentenced to death despite very little evidence.
Moreover, it isn't the Russian Revolution and Lenin that we see triggering people talking about the need to act. Bakunin in 1870 talks about how 'we must spread our principles, not with words but with deeds, for this is the most popular, the most potent, and the most irresistible form of propaganda.' This was quite a common theme in anarchist literature at the time, and even if they weren't explicitly anarchist, groups like the Molly Maguires took this message to heart with their violent attempts to gain trade union rights.
So to kind of sum this up; anarchism was an influential ideology long before the revolution, it attracted a lot of criticism and persecution from senior figures, and unlike other Leftist figures such as Debs it was an ideology that actively promoted violence. For these reasons it was seen as a significant threat.