Democratic officials said the White House wouldn’t have released the president’s long-form birth certificate if Mr. Obama’s job-approval ratings were strong and his personal favorability ratings were high enough to counter negative innuendos. But, they said, with so many economic factors souring the public mood, especially soaring gas prices, there was growing fear in the West Wing that angry Americans were more open to believing the conspiracy theories surrounding Mr. Obama’s eligibility for the presidency.

By releasing the fuller birth certificate, White House and Obama campaign officials were also hoping to take away—or at least tone down—Mr. Trump’s megaphone. One Democratic official said the attention showered on Mr. Trump was making likelier candidates for the GOP nomination, such as former Massachusetts Gov. Mitt Romney, look more acceptable by comparison. That, in turn, would make it harder to portray the eventual GOP nominee as catering to the conservative fringe.