When asked about what the time was where religious fervour and zeal reached its peak, most people would probably say "the Middle Ages". But, well - it's not quite right. It's hard to quantify religiosity, of course, but at least for the Catholic Church the heyday more likely has to be located in the Baroque, i.e. that era between the late 16th and the late 18th century (depending on where exactly you're looking) where the cultural and especially religious spheres were totally dominant in everyday life, made all the clearer by having a direct comparison to Protestant Europe where this was very much not the case.
What the Baroque also has going for it is that historians can generally work with much more sources than the Middle Ages. While we therefore also can't rule out that medieval Europe was teeming with monastic clergy much more than we thought before, we can at least say that there were a lot of monks and nuns in the Catholic Baroque, and at least to my knowledge there is no reason to assume that their relative numbers didn't eclipse all other periods in history.
So, how many men were living in monasteries during that time? Well, first there are the limits of what we can say: While the sources tend to be much better for the Early Modern Era than for earlier times, they are still far from being truly comprehensive, especially when it comes to demographics and reliable statistics. That said, some scholars have tried to gather as many data about early modern clerical demographics as possible.
As mentioned above, the Baroque's exact timeframe varies between countries and regions. Nevertheless it seems to be a safe bet that for all of Europe altogether, the peak of monastic religiosity was reached in the mid-1700s. Taking a look at the disparate statistics, we know for example that in 1792 about 0.26% of all Bavarians were men living in monasteries, but that was at a time when the Baroque boom had been long past. The Habsburgs census of 1781/87 came up with some 24,000 monastic clergy altogether in the dynasty's various territories, most of which were men. In France there might have been more than 3,000 male monasteries with an unknown number of total inhabitants. In 1765, the Portuguese authorities counted 531 monasteries with roughly 42,000 monks and nuns in between them at a ratio ofabout 3:1. For 1650 the inchiesta innocenziana, a monastic census ordered by Pope Innocence, came up with exactly 6,238 male convents in all of Italy with ~70,000 men living in them equalling 0.61% of the population, although later research shows that even this ambitious census wasn't fully comprehensive either. The Capuchins, one order out of many (though a pretty popular one, to be fair), rose to 33,000 members in 1754.
So, how many altogether? It's of course hard to say, but Peter Hersche estimates that at its peak up to half a million men and women might have lived in monasteries in Europe alone, with men making up at least half of that. For all of Christian Europe, Protestants and Orthodox included, this would come out as a a percentage of maybe 0.25% of men living in monasteries and quite possibly even more (depending on your exact definition of "Europe", of course).
Applied to all of Europe, this number isn't all that helpful however, at least if you ask me. The differences between the regions were vast; while in Protestant areas the number of monks was obviously zero, their percentage could easily rise up to 5% of the population in some mediterranean cities like Lecce and Valladolid.
Sources:
As a follow-up question: how does the percentage of people who have either taken vows and living in monasteries/convents along with lay people living there change when talking about Medieval Europe in comparison with other countries with strong monastic traditions such as the Theravada Buddhist kingdoms of Southeast Asia or the Tibetan Buddhist cultures of the Himalayas?