I did not play Witcher 1+2 and didn't read any of the books, so my reference pool is Witcher 3 only.
When I came to Kaer Morhen I was quite surprised to see that it is in such a bad shape. Most of the peripheral buildings in the surrounding are in ruins and the castle itself is in a state of disrepair. There appear to be only 3 witchers left who still live there and they seem to be no longer interested in training new recruits (in fact, the biography of Lambert mentions that he is the last one who was trained in Kaer Morhen).
What is the story reason why the Witcher caste is in such a decline and in which game/book did this happen? It certainly can not be lack of demand for their services: During his travels Gerald got jumped by customers left and right (or is that a case of gameplay-and-story-segregation and there are in fact less monsters at the time witcher 3 takes place than there used to?).