I was thinking particularly in the case of the high-water mark of their conquests, as they approached Vienna itself before they were eventually repulsed in 1683 and their European conquests began to be gradually lost.
I can imagine that in Vienna itself the mood would obviously be quite dire, but what about in other parts of Christendom? What did kings and nobles or peasants think in Rome, or Paris, or London, or throughout the rest of the Holy Roman Empire, if they thought anything at all? Was the Ottoman threat a distant one to them, something they probably just didn't see as likely to ever affect them? Or did any of them have great fear that they might be next in falling to the empire's conquests?
Okay, this will be my first time I comment and Im not sure am I qualified for that, but im currently reading History of Serbs by Vladimir Ćorović, which contains a lot of informations about relations of Ottomans and Europeans since Ottomans conquered Serbia, I dont know much about period later from 1562 so I will talk up to that point.. There was fear, but not strong enough to prevent Europeans from fighting each other. By the time Ottomans conquered most of the Balkans, Hungarians were next on the dinner menu, so they were mostly afraid, but Papal state, and Venice (which had some territories in Dalmatia) were also afraid of losing much of land and maybe even being fully destroyed. Ironicly they never stoped fighting each other(in Hungary there was a lot of fights about who will go to the throne after one Kings death), and Popes promised help of all Europe to defeat threat from east, but help rarely came. There was much of lords in destroyed balkan slavic states who accepted to continue to fight Ottomans because they were promised aid from western Europe and were left to die. So there was fear in central Europe and states who were directly threatened by Ottomans, but I never heard of fear of them further in Europe(France, England and so on). Hope I helped, If Im wrong in somethig please dont mind and let me know where I made mistake. source: History of Serbs by Vladimir Ćorović
France saw it as a way to reduce the influence of the Habsburg, the Genoans and the Venicians. They allied the Turk after the failed attempt to take Vienna in 1529. The alliance was made in 1536. They joined force to wage war of Italian city states and for the Hungarian campaign, in 1543. So in the case of France, i would say they did not saw it as a threat to them. They saw it as a way to destroy their enemies. That goes for the 16th century. Francis I and Charles V where bitter enemies. The ottoman pressure was part of the decline of the Holy Roman Empire. According to historian Arthur Hassall the consequences of the Franco-Ottoman alliance were far-reaching: "The Ottoman alliance had powerfully contributed to save France from the grasp of Charles V, it had certainly aided Protestantism in Germany, and from a French point of view, it had rescued the North German allies of Francis I."
So again, France saw the Ottoman as allies not enemies.
I would like to add a tack-on question.
I recently got the boardgame Here I Stand, which covers the first decades of the 1500s. There are six playable sides: Ottomans, Hapsburgs, England, France, Papacy, and Protestants.
In our games, the Protestant player has worked very closely with the Ottoman player to hamper the Hapsburgs/Papacy/Italians.
Did the Protestant powers have diplomatic relations with Suleiman the Magnificent? If so, what were these relations like? What level of coordination did they have in any wars against Charles V?
Since 1526 most of Hungary had been occupied by the Turks. The Hungarian plain had been a theater of warfare between the armies of Vienna and Constantinople for generations. In 1663, the battle flared up again, when the Ottoman armies started moving up the Danube. A mixed force from all Christendom obliged the Turks to accept a twenty-year truce a year later.
Louis XIV, the Sun King, enemy of the Holy Roman Empire, actually incited the Turks to resume their assaults, which they did in the end. So the French and the Ottomans were allies through common hostility to the Habsburgs.
Furthermore, during the Siege of Vienna the Ottoman army wasn't solely made up out of Muslims, there were Christians as well (Rumanian and Hungarian).
Concluding, it was a complicated situation and I don't think there was a sense of existential dread.
Source: A History of the Modern World, by R.R. Palmer.