Answer: Let us look at the proof text concerning blood used by one major church denomination to justify their teaching that donating it, or having a transfusion, is not God's will. It is found in Acts 15, where the apostle James is addressing a large group of church brethren (known as the Jerusalem Conference) gathered to discuss and debate whether circumcision was necessary for salvation.
James states to those in Jerusalem regarding Gentile converts, ". . . my judgment is that we do not trouble those of the Gentiles who have turned to God; But that we write to them to abstain from pollutions of idols . . . and from what has been strangled and from blood (Acts 15:19 - 20, HBFV). Basing church doctrine on a single scripture, especially in the case of donating blood which could save the life of others, is not a wise thing to do.
Note that the above verses are not discussing the donating or receiving of some bodily fluid. These verses are referring to a sometimes common practice among Gentiles (non-Jews) of eating or drinking blood. It was a practice that God forbid the Israelites, who had just left pagan Egypt, from doing when he said, "It shall be a statute forever for your generations throughout all your dwellings that you eat neither fat nor blood" (Leviticus 3:17).
Three out of the four items referenced in Acts 15:20 concern eating. God's word says to abstain from eating any of them for health and religious reasons. Sadly, though our Creator says not to do it, many people injest various forms of animal fluids like blood through frying or otherwise leaving it in meat for the "flavor" they offer!
The judgment rendered in Jerusalem concerns what to avoid in killing an animal for food or what parts should not be eaten by humans. It does not concern either donating or receiving blood from another human for the express purpose of saving their life.
The procedure of transfusions or the ability to donate and store blood for saving lives is, of course, not directly mentioned in the Bible. These medical advances have only been available in relatively modern times and were, of course, unknown to the ancients. These medical practices, however, are not any more of a sin than having an operation, or a root canal performed on a tooth and so on.
Here is a Biblical principle to consider regarding giving blood or receiving a transfusion. Hearing that Jesus had refuted the arguments of the Sadducees, the Pharisees got together to have their try at tripping him up. One Pharisee tested him by asking what he thought was the greatest commandment in God's law (Matthew 22:36).
Jesus responded by stating the greatest commandments were to love God with all your heart and soul and to love others like yourself (Matthew 22:37 - 40). He later raised this standard to the fullest when he told his disciples that they were to love each other as He (Jesus) loved them (John 13:34, 15:12)!
If your neighbor's life could be saved by donating your blood, or a transfusion of such fluid, does that not abide within what Jesus called the second greatest commandment? This certainly seems to be the case.