  We study the dynamics of Gaussian quantum steering in the background of a
Garfinkle-Horowitz-Strominger dilaton black hole. It is found that the gravity
induced by dilaton field will destroy the quantum steerability between the
inertial observer Alice and the observer Bob who hovers outside the event
horizon, while it generates steering-type quantum correlations between the
causally disconnected regions. Therefore, the observers can steer each other's
state by local measurements even though they are separated by the event
horizon. Unlike quantum entanglement in the dilaton spacetime, the quantum
steering experiences catastrophic behaviors such as "sudden death" and "sudden
birth" with increasing dilaton charge. In addition, the dilaton gravity
destroys the symmetry of Gaussian steering and the latter is always asymmetric
in the dilation spacetime. Interestingly, the attainment of maximal steering
asymmetry indicates the critical point between one-way and two-way steering for
the two-mode Gaussian state in the dilaton spacetime.
