When Robin, the Boy Wonder got introduced in 1940, it started off the kid sidekick trope in superhero comics and just as significantly, the move also gave young male readers a four-color reflection of themselves. Part of the character's creation might have been market-driven. With adult superheroes popping up all over the place, a kid crusader on a comic-book cover would stand out. Still, it's a bit weird to think that some people hate The Dark Knight's junior partner when he's been around almost as long as Batman himself. Surely, the longevity of the concept would quiet such haters, no?
But it hasn't. The most infamous example of Robin hate happened when more fans called into a 1-900 number and voted to have Jason Todd—the second person to wear the Robin costume—die after being badly beaten by the Joker. The margin of live-vs-die votes was close and that in and of itself stands as testament to a strain of Boy Wonder antipathy that had been brewing for a while, though to be fair a lot of it was probably Jason Todd-specific.
Superhero comics get called out a lot for being adolescent male power fantasies. Its pantheon is filled with characters who deal with trauma in ways that would be unhealthy in the real world and who use might to make right. Batman's a classic example of that argument, some would argue. Bruce Wayne could've moved on from the death of his parents without creating a macabre psychological construct that essentially preserves his grief for his entire life. Is Batman emotionally stunted, then? You could make a case for that.
Robin, however, punches out that logic. When Bill Finger, Jerry Robinson and Bob Kane had Bruce Wayne take a suddenly bereaved Dick Grayson under his wing, it introduced an element of empathy and family to Batman's evolution. After all, Batman could've been a hard-ass and left the tearful, traumatized boy acrobat at the circus. But the oath that Dick Grayson swears in the Batcave doesn't get the kid on the vengeance trail. It also pulls Bruce Wayne back from a darkness that would eat him alive.
Part of what we see in the Batman/Robin relationship is the ability to create family out of incredibly dire circumstances. It also gives him something to lose. If Batman dies in the course of battling evil, he's fulfilling the idea of a hero's ultimate sacrifice. In a way, it's what he supposed to do.
But if Robin dies while fighting crime, it's more tragic. He's younger and—by virtue of that and Batman's caretaking—hopefully less bound to the dark obsession that drives his mentor. He's also a brighter counterpoint to Batman's dour demeanor. So, if evil snuffs out the Boy Wonder, there goes Batman down into that abyss of grief again.
Critics who've tried to explain the anti-Robin sentiment have theorized that the Boy Wonder undermines the aspirational aspect of the Bat-mythos. Batman's only human and part of the appeal of the character is the implicit promise that you can become him if you train, study and brood enough. But if you're a kid reading about Robin, he's already more awesome than you'll ever be at the same age. And, it's even worse if you're an adult reading Robin because the same reasons get multiplied.
Yet, I'd argue that Robin's very existence means that Batman's not so far off the bend that people can't still care about him. In one way, it doesn't matter who's wearing the R-emblazoned red vest. They're all protégés and student in Bruce Wayne's obsessive crusade, interchangeable according to the needs of plot and subtext. But in another way, the specific presence of Dick Grayson, Jason Todd, Carrie Kelley, Stephanie Brown, Tim Drake or Damian Wayne changes things greatly. You get the child who'll honor a father figure in their own way or the truculent spawn who haunts the same father's memory. Maybe it's the well-adjusted kid who handles the pressure abnormally well, the eager-to-please offspring working with a distant dad, or the prodigy brat whose skills outstrip their maturity. But the thing that connects all of them is that they highlight the Dark Knight's humanity in a way that's not incredibly bleak.