  We consider the Anti-de Sitter space $\mathbb{H}^3_1$ equipped with
Berger-like metrics, that deform the standard metric of $\mathbb{H}^3_1$ in the
direction of the hyperbolic Hopf vector field. Helix surfaces are the ones
forming a constant angle with such vector field. After proving that these
surfaces have (any) constant Gaussian curvature, we achieve their explicit
local description in terms of a one-parameter family of isometries of the space
and some suitable curves. These curves turn out to be general helices, which
meet at a constant angle the fibers of the hyperbolic Hopf fibration.
