I believe there are two explanations that need to be given when answering this question.

First: opinions on the events occurring in France changed over time.

Given that the French Revolution spanned a decade (1789-1799), and the effects continued well into the 19th Century, the citizens of other nations changed their opinions about the revolution in France many times. For example, in the United States, Jeffersonian Republicans looked upon the early years of the revolution in France as directly related to the anti-monarchical and republican spirit of the American Revolution. However, following the execution of Louis XVI in 1793 and the ensuing "Reign of Terror," even supporters of the French Revolution in the country backed away from the radicalism that was developing.

Secondly: The merits of French Revolution were hotly debated in other countries.

In both the United States and Great Britain the events of the Revolution in France inspired egalitarian movements among sympathizers of the French revolutionaries. The Jacobite movement in England had only been defeated in England about a decade before the revolution in France began and there were some in England (though marginal) who sympathized with the French. The English, however, would declare war on Revolutionary France during both the War of the First Coalition and the War of the Second Coalition. In the United States, Federalists (men like Washington, Hamilton, Adams) detested the French Revolution because though theoretically it resembled revolutionary events in America, they believed the egalitarian demands made in "The Declaration of the Rights of Man" were far too radical and threatened the stability of the young United States. However, as stated before, at least early in the Revolution, some Americans sympathized with the revolutionaries and saw them as heirs of their own struggle. The young nation was brought dangerously close to war with England, however, during the Citizen Genet Affair, which kicked off a more than two decade struggle where the United States sought to assert itself on the world stage but often found itself caught between French and English ambitions.

Another interesting example is the revolution in the colony of Saint-Domingue. During the French Revolution, slaves led a rebellion on the island that was the crown jewel of French sugar production and eventually created an independent Haiti from the first, and only, successful large scale slave rebellion. Leaders of this revolution, notably Toussaint Louverture, were inspired by the events occurring in France. This rebellion had large ramifications for slaveowners in the Spanish, English, and French New World who feared similar rebellions among their slaves.

Easy answer: For the most part, both the English and the Americans saw the French Revolution as dangerous to political and social stability, though there are exceptions. In hindsight, with rise of Napoleon and his wars across the continent, many other nation-states across Europe would come to adopt English and American views.

Recently I read „The Oxford History of the French Revolution“ by William Doyle. As irishpatobie already pointed out, the initial opinion was mixed, with the US being sympathetic and Britain a little smug at seeing their rival brought low. Many British also expected France to become a constitutional monarchy, like the UK. After the Reign of Terror started, Britain was shocked and frantically tried to extinguish domestic revolutionaries. This was echoed in most other countries. A special case is Poland, which at the time took up some talking points of the Revolution. However, it did so for local reasons that were unique to Poland and not because they wanted to end Absolutism. This also gave the Russian response to the Revolution a special spin, as they eyed Polish territory.

Other countries happened to share land borders with Revolutionary France and thus they were not so lucky. In Belgium, the south-western parts of the HRE and in Italy, contact with France would follow a similar pattern: A French army would surprisingly defeat the professional army of the Coalition, occupy the region and topple the old order. The parts of the upper-class that are sympathetic to the Revolution give them a warm welcome. Maybe a sister republic is founded. Then the impoverished French troops start foraging, paying with nothing but worthless paper money (assignats), generally showing the local population that they are inferior to the Grande Nation and suppressing the Catholic Church. And all the while the war is disrupting trade. Most local people were not too fond of the French and in some regions, such as southern Italy, the common people rose up against the regime that claimed to have liberated them.