If he did at all? American Evangelical Christians seem to have a very particular dogma unrelated to the older, more European-originated branches of Christianity. While I am sure there exists a broad pantheon of influence, can anyone point to specific influences of Kierkegaard that may still be evident in today's belief system?
If you don't get someone posting a good answer to this, you should try it over at r/theology.
Fairly well-known is the fact that neo-orthodoxy’s reception of Kierkegaard influenced certain dimensions of evangelicalism. But see also, from the preface of Kierkegaard’s Influence on Theology: German Protestant Theology, ed. Stewart (p. x):
“The response of evangelicals to Kierkegaard was … strong and exceedingly diverse. The Reformed scholastic Francis Schaeffer (1912–84) caricatured Kierkegaard as a virulent irrationalist who undermined the objectivity of faith. On the other hand, John Edward Carnell (1919–67) cautiously applauded Kierkegaard’s focus on subjective appropriation but continued to critique his lack of interest in the rational justification of religious belief. More recently, Stanley Grenz (1950–2005) who likewise appreciated Kierkegaard’s theme of personal appropriation, seeing it as an antidote to fundamentalism, nonetheless critiqued Kierkegaard’s alleged neglect of the relationality of human nature.”
For more on this, see Kierkegaard’s Influence on Theology: Anglophone and Scandinavian Protestant Theology, also ed. by Stewart. In that volume, see especially Silas Morgan’s “Edward John Carnell: A Skeptical Neo-Evangelical Reading,” and Paul Martens’ “Stanley J. Grenz: An Unfinished Engagement with Kierkegaard.”
You might also find interesting theologian Roger Olson’s three-part series, “Was Kierkegaard an evangelical?” at Patheos (1, 2, 3).