If society decides the risk of these types of behaviors is significant they could pass a law making it illegal to ask someone how they voted. For those who would violate such a law even if it existed, they are by definition criminals, and already have the ability to tell you "Give me your banking passwords, or else". Blockchain itself won't cure all the problems with humans being criminals in this world, but blockchain itself DOES cure the problem of hacker-proofing voting data.

BTW: even if the blockchain data were totally held all within local governments only, it STILL addresses at least the problem of tampering at any level other than where the information is first collected. Even if the blockchain DB doesn't go as fine-grained as per individual it still "hardens the system" considerably from hackers, in a way that paper could never do, because recounts of paper is susceptible to human miscounts (both intentional and unintentional), and blockchain is absolutely not.