PETERBOROUGH, N.H.—Pete Buttigieg’s presidential campaign is brushing off heated criticism from liberals over everything from his college-tuition proposal to his health-care plans to his recent and past comments about race.

In interviews in the second state that will hold a Democratic nominating contest in 2020, many self-described liberal voters also said those critics didn’t bother them, emphasizing that they cared more about finding a candidate who could beat President Trump.

“Most...