In hindsight, they probably did not have much of a chance of success. Electoral Communism had its high-water mark on the Continent as minor coalition partners to much-farther-right, distinctly-anti-Soviet labor and social democratic parties in proportional representation nations. (It never had more than one or two MPs in first-past-the-post Britain.) Revolutionary Communism was DOA almost everywhere it wasn't imposed by Soviet occupiers. The closest Communism got to power in the West was Republican Spain, and that through a lot of clever backstabbing in the chaos of the Civil War, with the end result not of victory but of enabling Franco to mop up rather more easily than he might have against a unified center-left coalition.

The rather more interesting question is, with the evidence then available, were those fears reasonable. I think that here you would say, "somewhat so, at least."

Data points supporting this:

The Soviet revolution was truly shocking, a new thing on the face of the Earth. If the French Revolution and the Terror had been a passing fit, this was the chronic disease. It was not unreasonable to believe the Soviets would export revolution or that a sufficiently motivated Communist movement, supported by the Soviets, might powerfully try to execute something similar given a sufficient background of disorder. (Indeed, the Spanish Civil War proved those fears entirely right, even if the effort still fell short).

World War One had created death and dismemberment, industrialization, economic mobilization and privation, and social tumult quite unlike anything Britain, France and Germany had experienced. It was not at all unreasonable to expect worker militancy and political mobilization. And, indeed, that happened. Unionization exploded, political parties favoring or favored by unions grew dramatically, etc. Electoral and revolutionary Communism were not able to catch the benefit, but the property-owning establishment couldn't have known that more centrist parties and labor organizations would be the beneficiaries. (In the UK, Labour surpassed the Liberals first in 1922, and were in government, albeit short-lived, in 1923.) That the post-war environment would be generally fruitful for unions and their political parties was re-demonstrated in 1945 by Clement Attlee's Labour victories coming despite Churchill's personal popularity.

I cannot speak for the entire world, but in Germany Post-War Communism did not have any chance of actual success.

The November Revolution, from the outside, may sound like (as it was meant to) a Bolshevist styled uprising like in October in Russia, but the extreme-Left of the social-democratic party was a minority, uncertain of how to go about a revolution, but definitely wanting to avoid violence.

To avoid a lot of perhaps unecessary detail it can be framed this way: If the extreme-Left of the Social-Democratic party (which subsequently formed the German Communist Party KPD) was not only already a minority, but also ideologically split themselves, and if violence was to be avoided and the entire nation was shook to its core from the worst war in human history and a moderate party representing a good portion of the German civilian population and the officers of the army has most of the power, the KPD never really stood a real chance.

Most Western Governments at this point were already aggressive towards Bolshevism, meaning domestic leftist movements were stigmatized through association. Thus, with few exceptions, Communist Parties never really had much of a chance post WWI.

If you are curious I could be more specific, but I hope this helps!

Certainly the belief in the possibility that communists could come to power -- if not via the ballot box then via the bayonet -- was widespread. First a bit of background information (jump to the last two paragraphs for a tl;dr). Since around the turn of the century, it was clear that a split had emerged among social democrats regarding the means to best ameliorate the lives of the proletariat. "Social democracy" itself was a nebulous concept: for example, Bolsheviks and Mensheviks existed together, if uneasily, in the Russian Social Democratic Labor Party for a decade and a half, although using contemporary terms we would call the former "communist" and the latter "social democrats". In other words, Communists (Marxist-Leninists) advocated for the need for sudden revolutionary action, while Social Democrats (Marxists) had acquiesced to political reform via gradual, parliamentary change. (Although, based on responses to Eduard Bernstein during the Revisionism crisis beginning in 1896, social democrats were not willing to admit this themselves -- the German SPD did not formally abandon class struggle versus capitalism until the Godesberg Program of 1959).

A quick look at the progression of the Three Internationals showing the winnowing of socialist ideas shows how we got to the post-WWI environment: The First International (1864-76) contained a broad spectrum of leftist schools of thought, including anarchism and trade-unionism, and is best known for the struggle between Bakunin and Marx for intellectual leadership of the movement. The Second International (1889-1916) was marked the ascendancy of German Social Democracy in international socialism, in part due to its vibrant presence while outlawed (1878-1890) and impressive showing Reichstag elections. Though ultimately torn asunder by the inability of social patriots and anti-socialists to reconcile with one another over World War I, this period marked the high water point for "classic" international social democratic thought. The Third International (1919-1943), by contrast, featured explicitly communist parties as participants. Revolutionary violence was considered acceptable in the fight for human emancipation and world communism.

Thus in addition to the Russian Civil War and German examples already provided, one could look to the Finnish Civil War of 1918, the Hungarian Soviet Republic of 1919 and attempted coups in Georgia (May, 1920), Bulgaria (September, 1923) and Estonia (December, 1924). Except for the victory of the Bolsheviks, none of these Reds ultimately prevailed. The next time, however, they might. It was in this sense that the specter of communism haunted the minds of many in the immediate post-War years and led to widespread red-baiting throughout the interwar period.

It is sometimes difficult to appreciate (then and now) in Western Europe and North America how notoriously unstable parliamentary democracy was in the newly independent states of Central and Eastern Europe. Their fortuitous emergence from the ashes of the continent's empires meant that they were unprepared for the challenge of democratic governance, not to mention foreign policy questions regarding border and extra-territorial co-nationals, and thus most vulnerable to radical political change -- and not just communist. In fact, if the years through 1925 were marked by communist coups, then the years 1926-1939 (beginning with Poland's May Coup) were marked by authoritarian counterreactions, often "justified" by the putative threat of further communist unrest. (The First Czechoslovak Republic and Finland are notable exceptions to this regional trend.)