For children, the reason to hate Barney was... well, he looked stupid and he sang about how much he loved you. It was the 90s... it was the decade where it was okay to be openly cynical and even homophobic. I'm not defending the behavior, just pointing out the attitude.
On a more sociological front, Barney represented a change in children's educational programming from being "educational AND entertaining" to "entertaining with education as a secondary concern". Before then, people who wanted to create kid's programs were often stuck with the quandary of having to teach audiences practical lessons and things they'll eventually need to understand when they go to school while balancing their creative ambitions. The standard bearer was, of course, "Sesame Street", which for nearly two decades, had proven it was possible to combine both thanks to intensive studying of young audiences and their mannerisms. Barney seemed to be the first, however, to disregard the practice of pleasing both masters and finally allowed wannabe thespians to do whatever they wished provided they did token lessons and stuff. Even so, Barney often serviced as a bad role model, where in spite of the fact he was supposed to be the childlike creature in theory, in execution he was the adult, so seeing Barney have kids disturb wildlife, take things that aren't theirs, and be disrespectful to privacy sent a poor message. Some people joke about moral guardians nowadays or hold them in contempt, but considering the popularity of Peppa Pig and how long freaking Caillou lasted in syndication on PBS, with both shows serving as extremely poor examples of behavior and guidance for impressionable youth, it's not too farfetched to stop and think about the consequences of a purple dumbass going "Super-Dee-Duper!"
As for the open hatred amongst older folks? I assume they took a look at Barney and went "What's so special?" and decided it was a very easy target, which given how litigious the company that owned the franchise was at parodies and satire, made people turn around and hate Barney even more given that they only seemed to care about a profit motive instead of anything else. I mean, anyone would want to protect their intellectual property, but it seemed like they specifically had it against anti-Barney humor.