Here is a very comprehensive article about The Law and Blood.
Quote from the article:
It is still true that blood is sacred; Jesus died to save us from our sins, and his blood is spoken of throughout the NT as vital and sacred. The REAL question is, is the eating of blood still to be equated with profaning life? Not in most modern cases of incidental consumption of blood. Furthermore, does the usage of blood as a vehicle of saving lives (as in blood transfusions) at the same time constitute a profanation of life, or a respect for life? The answer should be obvious.
It also must be clarified that transfusing blood is not the same as eating blood. The WT confuses the issue by pointing out that blood acts as a kind of "food" in transfusions (see Blood, Medicine and the Law of God, pages 17,18), whereas the REAL issue they should be concerned with is whether or not the use of blood to save a life is profaning it. Such is certainly not the case when a person is dying, and a blood transfusion may save their life.
It is important to note that orthodox Jews today, while STILL OBEYING the OT laws against the eating of blood, believe there is nothing wrong with taking a blood transfusion. Even the legalistic Jewish mind does not make such a connection, yet the Governing Body of the WT forces this interpretation on their people.
Much more at the link above.