Being "affine" in this case does not make much sense, because the hyperkaehler deformation is a complex manifold, without a fixed algebraic structure. Simpson produced an example of a hyperkaehler deformation of a space of flat bundles admitting several algebraic structures, both inducing the same Stein complex structure; one of them is affine, another has no global algebraic functions. In fact, the space F of flat line bundles on elliptic curve (with an appropriate algebraic structure, defined by Simpson) is an example of such a manifold, it is biholomorphic to $C^*\times C^*$, but this biholomorphic equivalence is not algebraic, and F has no global algebraic functions.

However, you can show that a hyperkaehler deformation of a resolution of something affine has no non-trivial complex subvarieties (arXiv:math/0312520), except, possibly, some hyperkaehler subvarieties The latter don't exist, because the holomorphic symplectic form $\Omega$ on such a manifold is is lifted from the base, which is affine, hence $\Omega$ vanishes on all complex subvarieties.

Therefore, a typical fiber of such a deformation is Stein.

Indeed, a hyperkaehler deformation of a resolution of something affine remains holomorphically convex. To see this if you produce a function which is strictly plurisubharmonic outside of a compact set (we have such a function, because we started from something affine), and apply the Remmert reduction.