I believe that the use of biodiesel is safe. I do not see any reason to think otherwise. I think that biodiesel has been around and used for a long time without any negative side effects. I think that until there are significant evidence and proof to say otherwise, the use of biodiesel is safe.
Biodiesel is as safe as regular diesel, with the added benefit of being better for the environment. Producing diesel from fossil fuels removes carbon from sequestration in the earth, releasing it into the atmosphere once burned. Biodiesel is produced from vegetable oils and/or animal fats. As a plant grows, it sequesters carbon from the atmosphere. When burned as biodiesel, it is returned to the atmosphere. So no additional carbon is released, while petroleum based diesel releases additional carbon which would have been sequestered underground.
Biodiesel works the same as gas: creating a mini explosion in an engine so that the pistons move and the motor can turn an axle. Burning dead organisms is just a terribly inefficient process and produces a lot of undesirable gases.There's little way around it except for nuclear and hydrogen and those carry different risks that many people aren't willing to accept. Electricity is the cleanest option for the consumer, but batteries still aren't all that enduring and it just means that the electric company burns the dead organism instead of the individual. So, biodiesel is a pretty good alternative to gasoline if for no other reason than that it is hard to monopolize on. It should be cheaper. It is also easier to produce, renewable, and can be made from otherwise useless biological wastes as well as specially grown plants and algae.
Biodiesel is safe. Well, it is safe when compared to many of the other forms of fuel. The way it is made is better for the environment, and the emissions it emits are safer for the environment than other fuels. Is it perfect? No. But it is certainly a step up.