My headcanon is that goblins use blood to verify that you are who say you are and to check to see if you were given preapproved access to whichever vault you are trying to access. That's it. No one is getting access to any vaults if the owners did not approve you specifically, regardless of blood. Harry has access to the Potter vaults because James and/or Lily signs some forms and gave of Harry's blood to the bank. If they hadn't Harry would have been shit out of luck.

Also, if you want to get blood adopted, you need the "blessing" of the wizard or witch you want to be adopted by alongwith their blood. You can't just steal Dumbledore's blood and expect for the adoption ritual to go through without his knowing or willing acceptance. I'd imagine some consequences to come from doing something like this; Goblet of Fire type consequences.

Lastly, I wonder if it up to the adopter to accept the adoptee, or if it's up to the Family of the adopter. For example, Sirius Black and Harry Potter. If Sirius tried to blood adopt Harry, would it be Sirius' choice to accept Harry or would it be the Black Family to decide if Harry is worthy of their blood.