With higher approval ratings from critics and audiences? Unlikely.
Whatever ultimately prove to be the reasons for the lower box office of Beyond, a dislike for Into Darkness will be minuscule among them. For the general audience (FAR more important than "fans" in terms of box office success), the various things "fans" (some, certainly not all, nor even most) find problematic with Into Darkness are not even blips on the screen. General audiences didn't care that Cumberbatch didn't look/sound like Montalban; didn't care the Klingons looked different; didn't care about how quickly the Enterprise made it to the Klingon home world; or about transwarp beaming; or "magic blood"; or... None of that. They enjoyed a visually stunning, exciting action movie and moved on.
Uneven marketing, poorly chosen release date, unusual movie going pattern this summer, three years since last film, a further shift towards waiting for home access to movies, aging demographic of Trek audience (ties into home release trend--a growing issue for all studios and films)--each of these factors is WAY more significant than a negative view of Into Darkness.