More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe.
On the other hand, Silver (1971) showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees.
The existence of an inaccessible cardinal is in fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then the cardinal ω2 is inaccessible in the constructible universe.
The existence of a binary κ-Kurepa tree is equivalent to the existence of a Kurepa family: a set of more than κ subsets of κ such that their intersections with any infinite ordinal α<κ form a set of cardinality at most α.
The Kurepa hypothesis is false if κ is an ineffable cardinal, and conversely Jensen showed that in the constructible universe for any uncountable regular cardinal κ there is a κ-Kurepa tree unless κ is ineffable.