Not really no. Most blood transfusions we think about are red blood cells or platelets, which don't have the immune function you're asking for. That's a good thing. Usually, if there are white blood cells in the transfused blood, the host's immune system will recognize them as foreign and destroy them. Remember, your cells all look like foreign invaders to my cells; blood transfusions of red blood cells are carefully matched to limit negative reactions. There is also a process called transfusion-associated graft versus host disease in which the donor white blood cells will attack the host cells; this mainly occurs in immune-compromised individuals, but GvHD is definitely something to avoid. Blood transfusions are usually filtered and irradiated to remove, among other things, white blood cells.
That being said, people are beginning to use white blood cells as treatment. A new therapy being studied heavily for all sorts of diseases, from cancer to HIV, is to take the hosts own white blood cells and grow them up in the lab to select for the strongest and most effective cells. The researchers then wipe out the individual's immune system and give them a dose of their own, super-powered white blood cells, hoping that works.
Sometimes it kind of does. They've also been trying a new system, similar to what you propose, using bone marrow. They had a huge success with an HIV-positive individual now referred to as "the Berlin patient." They gave him a marrow transfusion which would produce HIV-immune white blood cells and replaced his immune system. He was and is effectively cured of HIV/AIDS.