It depends on the extent of the injury
I worked out with two torn rotator cuff muscles while waiting for my surgery, I trained everything else and paid extra attention to my motions so as not to aggravate my shoulder
I suggest getting checked out by your doc and a Physical Therapist, then abide by their guidelines.
in a case like that, I tend to incorporate the exercises from physical therapy into the workout routine but do not do any other exercise that involve the shoulder unless cleared to do so. I am not sure whether you are asking for yourself or for a client. Sometimes, you can aggravate a rotator cuff injury by the way you need to adjust machines, if using, so this is another area to be careful about.
Cardio is fine as long as it does not involve the arms. Even the arm movement of the elliptical can make matters worse, even though you may only notice it afterwards.