/u/Aethelric provides an answer to this question using a historical definition of folklore as being a system of older beliefs. That is an interesting approach and is probably the answer the OP was seeking. For the record - and defending the point of view of the folklorist - I will add that folklorists would answer this simply, that folklore never stops. Every culture has oral tradition, and in general, people have beliefs that subsequent generations may regard as "superstition" (setting aside those who are complete atheists; i.e. those who have discarded all beliefs in everything supernatural). Folklore changes and adapts, but it does not cease to exist. I trust that I'm right - knock on wood (if you catch my meaning).

This is a fairly broad answer, but I'll try to answer succinctly. The shortest answer is "depends highly on where, who, and when, and also upon your definition of 'folklore'." I'm giving a very general (and perhaps cavalier) narrative here, which works with the understanding of "folklore" to mean "traditional spiritual beliefs, often pre-Christian in origin or at least not found within the Biblical canon". Basically, "folklore" fell out of favor beginning primarily in the sixteenth century, and by end of the eighteenth century was largely "superstition" in Western/Central Europe.

The narrative: in the medieval period, people of nearly all classes and status maintained a similar epistemology, and roughly similar "folklore". This epistemology gave very high value to the explanatory power of supernatural forces. Supernatural beings of strange power are imminent in the medieval (and early modern) mind, and creatures and beings of all kinds of magical features exist throughout the world. For the peasant, these creatures exist right within the boundaries of the nearest woods, and have some capability to interact with the physical world. Peasants and townsfolk alike would engage in rituals and festivals which were believed to communicate with supernatural forces.

Enter the early modern period, and the Reformation. While the worldview of nearly everyone is still highly "magical", church officials (most strongly among Protestant churches, but also in the reforming Catholic Church) in the sixteenth century begin to excise certain non-Biblical or otherwise unsavory elements from their understanding of the spiritual world. This process is fairly innocuous and primarily urban, and proliferates among the educated classes. Towns, particularly in Reformed territories, begin to cease many of their former rituals and festivals, or at least tone down their "folklore-ish" elements. Men slowly become more associated with a system of knowledge more resembling what we would call "scientific" or empirical, although even the most successful practitioners of science are still highly invested in the supernatural world (Isaac Newton, for instance, famously wrote as much or more on alchemy and theology than "science"). Women, with less access to education, become more associated with traditional ways, from medicine to religious practice. Partially a side effect of this process, witch hunts begin that stop the open practice of traditional folk magic by rural people, alongside education efforts that seek to end them. As an aside, note that the gendering of science and magic to male and female, respectively, is still strongly felt throughout the West today.

It's worth noting, however, that even during this period, nearly everyone still accepted the potential power of the rituals, dances, and festivals. However, the understanding of the tenor changed for the educated: these activities consorted with Satan and demons, who tricked people into worshiping them by giving worldly favors. As a fairly inevitable result, "folklore" became increasingly out-of-favor (at least among the educated and the elite).