This is answered for free groups in Membership to double cosets in free groups and the same method basically works for hyperbolic groups. You might as well do double cosets $HgK$ with $H,K$ both quasiconvex. Then $w\in HgK$ if and only if $Hw\cap gK$ intersect nontrivially.
Now the set of geodesic words belonging to $Hw$ is a regular language can there are known algorithms for constructing an automaton recognizing these languages (HJRW mentioned this in the comments for subgroups, but it is more or less straightforward to generalize for cosets) and similarly for $gK$. Therefore the geodesic words in $Hw\cap gK$ are recognized by a finite automaton that you can construct. Since emptiness is decidable for finite automata, you are done. You need the quasi-convexity constants of course to build these automata.