By itself, not really serious. I guess it depends on your definition of " serious". The problem with bad engine mount is this: the vibration that is not dampened by the mount will now travel to other parts of the engine and will a broken mount will affect other mounts.
Depending on which mount that is bad. If the front or rear, now your engine will " flex" forward and back with every acceleration and deceleration. This create stress to your exhaust system. If it is the dog-bone (front engine - by the belts), the engine will flex side to side, which will transfer stress to transmission.