Think there was a lot of skepticism towards Cardinal Rodrigo Borgia on account of the fact he had many children. However, it wasn’t that unusual for cardinals and even Popes to have fathered children even though they were men of the clergy. Pope Alexander VI was by no means the first the pope who had sons and daughters. His predecessor Pope Innocent had also fathered children prior to becoming Pope. Being a clergy member and having children is not really something that goes hand in hand, but it came to be accepted or not regarded as important if the clergy was of “good faith.” But there were many who strictly opposed such behavior and thought being a priest or cardinal was not compatible with being a father to children. So I think Alexander having children certainly made him enemies and created differences with rivals, still it is likely a lot of his rivals just didn’t like him regardless of him fathering children, but the fact that he had so many children made them despise him even more. Alexander was notorious for being a womanizer, by no means the only one, he was someone who had a lot of sex while being a cardinal and also when he was Pope.
When it comes to Lucrezia and Cesare in the matter of incest, there is no clear evidence of such behavior. Most of these accusations come from Pope Alexander’s successor and rival who was former cardinal Giuliano Della Rovere, Pope Julius II. Julius despised Pope Alexander, and is the primary person who has made the assertions that Alexander, Cesare and Lucrezia was involved in a triangle affair. It has to be said, there were many rumors going around about a possible incest affair between Lucrezia and Cesare. Where there is smoke there is fire they say. It is impossible to know for sure, what we do know is that the Borgias were hated by a lot of people and rumors like this might have been made up or exaggerated. Still, if there ever was a relationship beyond brother and sister between Cesare and Lucrezia I find it very hard to believe Alexander was involved. After all he loved woman, that is well known, but having sex with his daughter I think is a step too far for even him. This is a general overview, there are a few articles out there that might give a better description of the notorious Borgias of the Renaissance.
I suggest reading this article http://www.historytoday.com/blog/2013/10/were-borgias-really-so-bad
And maybe listen to this podcast:
Follow up question: I've heard that some historians question whether Rodrigo Borgia was actually the father of Lucrezia and Cesare (as opposed to an uncle), and if the rumored parentage was accepted as fact due to the unpopularity of the Borgias by the other powerful Italian families. Is the parentage generally accepted to be true, or is the potentially another Borgia rumor lost to history?