As Hugh Nibley explained in 1952, in an article printed in the official publication of the Church at the time, the Book of Mormon identified Asia as a source for ancient Native Americans long before anthropologists did. The essay was "The World of the Jaredites," Improvement Era, Vol. 55, June 1952, from which I quote:
That account [the Book of Ether in the Book of Mormon] tells us that at the very dawn of history, many thousands of years ago, a party of nomad hunters and stock raisers from west central Asia crossed the water--very probably the North Pacific--to the New World, where they preserved the ways of their ancestors, including certain savage and degenerate practices, and carried on a free and open type of steppe warfare with true Asiatic cruelty and ferocity; it tells us that these people moved about much in the wilderness, for all they built imposing cities, and that they produced a steady trickle of "outcasts" through the centuries. A careful study of the motions of the Jaredites, Mulekites, Nephites, and Lamanites should correct the absurd oversimplification by which the Book of Mormon as a history is always judged. It will show as plain as day that the Book of Mormon itself first suggests the Asiatic origin of some elements at least of the Indian race and culture long before the anthropologists got around to it. The scientists no longer hold that one migration and one route can explain everything about the Indians. The Book of Mormon never did propound a doctrine so naive. Though it comes to us as a digest and an abridgment, stripped and streamlined, it is still as intricate and complex a history as you can find; and in its involved and tragic pages nothing is more challenging than the sinister presence of those fierce and bloody-minded "Men out of Asia" known in their day as Jaredites....
I think by now it should be apparent that the Book of Mormon account is not as simple as it seems. Ether alone introduces a formidable list of possibilities, few of which have ever been seriously considered. Foremost among these is the probability, amounting almost to certainty, that numerous Jaredites survived in out-of-the-way places of the north to perpetuate a strong Asiatic element in the culture and blood of the American Indian.[emphasis mine]
Thus, given that the apparently Asiatic Jaredites were on the continent long before the Nephites, and given that other migrations from Asia are permitted by the Book of Mormon, finding evidence of mostly Asiatic genes in the Americas does not necessarily pose a problem for the Book of Mormon. This understanding of the Book of Mormon (the Jaredites as an Asiatic migration, and the possibility of other migrations from Asia being allowed by the Book of Mormon) is not one just recently concocted to deal with recent DNA evidence--it was printed in the official Church periodical decades before critics used DNA evidence to attack a common misreading of the Book of Mormon. In fact, even if we were to erroneously conclude that the ONLY ancient migrations to the New World are those described in the Book of Mormon, the heavy presence of Asian genes in Native Americans could still be compatible with the apparently Asian origins of the ancient Jaredites, whose descendants may have spread across the continent and obviously were present in Book of Mormon lands in Mesoamerica even after Ether saw their central groups wiped out in a bloody civil war.
See my LDSFAQ page on the issue of DNA and the Book of Mormon for further details.
In Bob Bennet's surprisingly good and highly readable book about the Book of Mormon, Leap of Faith: Confronting the Origins of the Book of Mormon, he quotes the above passage from Nibley and further argues that the Book of Mormon should be given credit for pointing to an ancient Asian link in the gene pool of the Americas long before science established that connection. Interesting.
Bennet also makes the point that while the story of the Jaredites plays an important and pervasive background role in the Book of Mormon, the Book of Ether itself makes little sense from the perspective of a forger trying to craft something that will sell. All risk and difficulty with little to gain--would have been much better and more logical for a forger to just leave that out and stick with more familiar topics and themes. It's boring, dry, highly condensed, sketchy, and utterly different from the rest of the text in terms of culture and behaviors. For careful readers of the Book of Mormon, though, it plays a vital role and adds subtlety and dimensions of meaning that pervade the rest of the text. One example is the recently noticed relationship between ancient Jaredite names and later rebels within the Nephite people, suggesting that indigenous remnants of Jaredite culture brought in under Nephite rule were important sources for political and religious rebels like Corianton. Again, interesting. One of those subtleties that make sense if the Book of Mormon is an authentic ancient record that is, after all, "smarter" than Joseph Smith.
Oops, I diverged from the topic again. OK, back to DNA. If you've got some, be grateful. And if you or anyone else like, say, a Native American friend, has some Asian DNA, again, be grateful. It's great stuff and is no reason to let your faith be shaken up.