Stationary objects experience only the gravitational force. Such objects can be balanced only when the line of action (line connecting the centre of gravity of the object towards the earth) touches the ground at a point within the base of the object. If the base area is large, slight deviations due to external forces such as the wind will not push this point out of the base.
So it is easy to balance such objects. In the case of a two wheeler, the base is very narrow, almost a line connecting the two points where the two wheels touch the ground. Hence any slight disturbance will push the line of action outside the base area leading to difficulty in balancing.
During motion the situation is entirely different. In addition to gravity there is another force exerted by the engine. It acts in the forward direction.
The resultant of these two forces is in the forward direction and so the vehicle moves forward independent of where the line of action cuts the ground. Hence it is easier to balance a moving vehicle.