In any situation, responsibility and control can be either internal or external.
Internal responsibility and control:
For example, when dealing with your feelings, and internal sense of control means you feel that you control whether you are happy or sad. You find that other people don't really "make" you upset. An internal sense of responsibility means you feel it's your own job to be happy. You don't lay that responsibility on someone else.
External responsibility and control:
An external sense of control means that you think the world yanks your feelings around and you can only respond to outside events. An external sense of responsibility means that you think it's someone else's job to make you happy.
The healthiest situation is when both of these are internal—at least in regard to your feelings. In regard to someone else's feelings, these should both be external (to you). This means recognizing that their feelings are under their control, and that it's their responsibility to feel happy.
If you try to take responsibility for someone else's feelings, they won't give up control—but they might be glad to let you have responsibility. It will end up making you frustrated, sad, and anxious. In fact, any time you take responsibility for something over which you have no control, these are the likely outcomes.
To correct this you might try to grab control as well as responsibility, to MAKE them feel better. But it's unlikely they will give up control. Better (but still hard) is to recognize this dichotomy and to then give up responsibility (for their feelings).
You'll end up less frustrated and less burdened as well.