This position is most prevalent within the Independent Baptist branch of the Baptist movement. The rejection is based in part on the different texts which were used to translate the different translations of the Bible. Most modern translations are translated from the Alexandrian manuscripts, while the King James Version was translated from the Textus Receptus, or Received Text.
There are variations within the King James Only Movement. For example, the late John R. Rice, who published The Sword of the Lord, took a position that only the original Greek and Hebrew manuscripts are inspired scripture, and that all translations of those done in good faith are useful as scripture, but he expressed a preference toward the King James Version for aesthetic reasons. On the other extreme can be found the teachings of Peter Ruckman, who believes that the King James translation constitutes an "advanced revelation" from God which is superior to even the original Greek and Hebrew texts. Most King James Only advocates hold to a position somewhere between those two extremes.
The roots of the King James Only Movement can be found in the controversy over the publication of the Revised Standard Version (RSV) of the Bible in 1952, which was issued by the National Council of Churches (NCC). Many fundamentalists believed that the NCC was a hotbed of liberal theology or "modernism" and were suspicious of the new translation. Accusations of Communist and Vatican influence within the NCC were brought up, and fundamentalists largely rejected the RSV, although for three decades it became the most widely used Bible translation within the mainline and liberal Protestant denominations. One particular criticism of the RSV centered around the decision made by the translators to translate a number of Old Testament prophecies, which some scholars believed referred to the coming of Christ, in a neutral fashion which did not necessarily imply any connection to Christ. As a result, critics charged that the NCC, in issuing the RSV, had deliberately set out to discredit doctrines such as the virgin birth.
The King James Only Movement as we know it today began to take form after conservative and evangelical Christian groups began producing their own modern Bible translations, including the New American Standard Bible, the Good News Translation, and the New International Version. Most evangelicals who were wary of the RSV readily accepted these other new translations, but some fundamentalists did not. Those who rejected all modern translations began to formulate the doctrines which are held by the King James Only Movement, such as their belief that the Received Text is superior to the Alexandrian manuscripts. The King James Only Movement became one of the core beliefs within the growing Independent Fundamental branch of Baptists.
Within broader evangelical circles, the King James Only belief is controversial and is widely rejected. Most evangelical scholars, in fact, hold that the Textus Receptus manuscripts which the KJV was translated from contain a number of errors, and that the modern translations are translated from the earliest and most accurate manuscript evidence which we currently have.
Besides Independent Baptists, there are a number of other denominations which hold, to varying degrees, to a King James Only position. These include the United Pentecostal Church, the Church of Jesus Christ of Latter-Day Saints, also referred to as Mormons (who typically only use the KJV in the United States), and some of the small, conservative splinter denominations from the Episcopal Church USA which collectively refer to themselves the Continuing Anglican movement.