From what I have read, DNA testing has turned up zero evidence of any Khazar ancestry among modern Ashkenazi Jews, but plenty of evidence connecting them with the southern Levant--the Holy Land.
Some might assume that Koestler’s “descended from the Khazars” theory was based on some sort of animus against Jews but in fact he intended it as a counter to racial anti-Semitism. Koestler himself was a non-practicing English Jew, born in Hungary. His two autobiographical volumes are very good.