A gyroscope is not suitable. Maybe a reaction wheel could be an idea.
To explain why a gyroscope would not be suitable let me describe a situation where gyros work well: stabilizing a ship.
In the luxury yacht market you see usage of a design called control moment gyroscope.
Depending on the orientation of the ship with respect to the waves the ship will tend to have rolling motion. In order to exert a force a control moment gyroscope must have some leverage. On a yacht that is achieved as follows. When the control motion gyro counters roll to the right the bow of the ship is pushed down; when the control motion gyro counters roll to the right the stern of the ship is pushed down. (Or vice versa, depending on direction of gyro spin.) On a ship that pitching effect is negligably small; a ship is far more stable around its pitch axis than around its roll axis simply because a ship is longer than it is wide.
Going back to the case of hoisting up to a helicopter:
A single gyroscope can counteract tendency to spin, but you end up trading one problem for another.
Incidentally, another option, I suppose, is to outfit the basket with a small rotor, a rotor like the tail rotor of the helicopter itself. As with a reaction wheel design the basket tail rotor would have to be controlled by a sensor that senses that the basket starts spinning.
A reaction wheel design will probably be heavier than a basket tail rotor design.
Let's say that the wash of the helicopter does tend to make the basket spin. That is a continuous torque. To counteract that for an extended time the reaction wheel would have to keep increasing its angular velocity. Pretty soon the reaction wheel will max out, and then you have a fast spinning wheel on your hand. When the rescuers start handling the basket they will then have to deal with the counter-intuitive gyroscopic effect of the reaction wheel.
Conversely, with a basket tail rotor design: when the basket no longer tends to spin the basket tail rotor also stops doing anything.