After listening to Gov. Scott Walker of Wisconsin as he has traveled the country preparing his campaign for president, which officially begins on Monday, admiring voters most often describe him as “authentic,” “real” and “approachable,” Mr. Walker’s advisers say.

Two words these voters do not use about him? “Smart” and “sophisticated.”

“Scott is working on that,” said Ed Goeas, a veteran Republican pollster and a senior adviser to Mr. Walker. “Look, ‘approachable’ is worth its weight in gold in politics. ‘Smart’ is something voters look for in legislators who craft policy. But Scott is preparing hard to talk about every issue.”

As Mr. Walker becomes the 15th prominent Republican to enter the 2016 race, the crucial question he must answer is whether he can cross the threshold of credibility so that someone entering a voting booth can imagine him as president, according to several leading Republicans and interviews with regular voters.

While Mr. Walker is ahead in some opinion polls, including for Iowa’s first-in-the-nation caucuses, a series of early gaffes alarmed party leaders and donors and led Mr. Walker to begin several months of policy tutorials. The collective hope is that Mr. Walker can avoid what Mr. Goeas and other advisers describe as Sarah Palin’s problem — becoming a candidate who is initially popular among Republicans, like the 2008 vice-presidential nominee, but loses luster because of missteps as the campaign goes on.