It is certain that Joseph Smith was influenced by Kabbalah.
In his 1992 Book The American Religion: The Emergence of the Post-Christian Nation Harold Bloom writes:
What is clear is that Smith and his apostles restated what Moshe Idel, our great living scholar of Kabbalah, persuades me was the archaic or original Jewish religion. . . . My observation certainly does find enormous validity in Smith's imaginative recapture of crucial elements, elements evaded by normative Judaism and by the Church after it. The God of Joseph Smith is a daring revival of the God of some of the Kabbalists and Gnostics, prophetic sages who, like Smith himself, asserted that they had returned to the true religion. . . . Either there was a more direct Kabbalistic influence upon Smith than we know, or, far more likely, his genius reinvented Kabbalah in the effort necessary to restore archaic Judaism.
Unlike Bloom, this answerer is less optimistic that the similarities between Smith's writings and the Kabbalists were spontaneous, especially given that it is very well known that Smith's family was active in the Order along with the wealth of influence that Kabbalism and Hermeticism had on Masonic practices and philosophy.
In The Mormon Church and Freemasonry (2001), Terry Chateau states of the Smith family:
[The family] was a Masonic family which lived by and practiced the estimable and admirable tenets of Freemasonry. The father, Joseph Smith, Sr., was a documented member in upstate New York. He was raised to the degree of Master Mason on May 7, 1818 in Ontario Lodge No. 23 of Canandaigua, New York. An older son, Hyrum Smith, was a member of Mount Moriah Lodge No. 112, Palmyra New York.
In his article Joseph Smith and Kabbalah: The Occult Connection, published in 1994 and named the best article in Mormon studies by the Mormon History Association the following year, Lance S. Owens says,
While I would not diminish the inventive genius of Joseph Smith, careful reevaluation of historical data suggests there is both a poetic and an unsuspected factual substance to Bloom's thesis. Though yet little understood, from Joseph's adolescent years forward he had repeated, sometime intimate and arguably influential associations with distant legacies of Gnosticism conveyed by Kabbalah and Hermeticism—traditions intertwined in the Renaissance and nurtured through the reformative religious aspirations of three subsequent centuries. Though any sympathy Joseph held for old heresy was perhaps intrinsic to his nature rather than bred by association, the associations did exist. And they hold a rich context of meanings. Of course, the relative import of these interactions in Joseph Smith's history will remain problematic for historians; efforts to revision the Prophet in their light—or to reevaluate our methodology of understanding his history—may evoke a violently response from traditionalists. Nonetheless, these is substantial documentary evidence, material unexplored by Bloom or Mormon historians generally, supporting a much more direct Kabbalistic and Hermetic influences upon Smith and his doctrine of God than has previously been considered possible.
Through his associations with ceremonial magic as a young treasure seer, Smith contacted symbols and lore taken directly from Kabbalah. In his prophetic translation of sacred writ, his hermeneutic method was in nature Kabbalistic. With his initiation into Masonry, he entered a tradition born of the Hermetic-Kabbalistic tradition. These associations culminated in Nauvoo, the period of his most important doctrinal and ritual innovations. During these last years, he enjoyed friendship with a European Jew well-versed in the standard Kabbalistic works and possibly possessing in Nauvoo an extraordinary collection of Kabbalistic books and manuscripts. By 1844 Smith not only was cognizant of Kabbalah, but enlisted theosophic concepts taken directly from its principal text in his most important doctrinal sermon, the "King Follett Discourse.".