Whether voting via the Internet could be made "secure" depends upon what is meant by the term. Historically, a key aspect of the secret ballot is that it should not only be impossible for someone to determine how an individual voted without that person's consent, but it should even be impossible to coerce or entice a person into proving how he or she voted. Obviously nothing can prevent a person from saying how he or she voted, but a person could claim to have voted for candidate who received at least one vote, regardless of how the person actually voted. Voting schemes which do not require people to cast votes under supervised conditions cannot very well meet the latter condition; if such supervision is regarded as security requirement, then such schemes cannot be considered "secure".
Further, even if the aforementioned requirement is not considered absolute, a fundamental difference between voting and banking is that a bank is supposed to know exactly who is doing what; by contrast, election officials are supposed to have no way of knowing or finding out who voted for whom. Thus, the sorts of reconciliation that are possible with on-line banking aren't possible with on-line voting. If thieves steal a person's credit card number, it's possible to determine which transactions were performed by the thief and reverse them. By contrast, if a voter's ID gets stolen and is used by someone else to cast a fraudulent ballot, the only way to reverse the fraudulent vote would be for election officials be able to determine for whom they were cast, which would clearly be contrary to any definition of "secret ballot".
Finally, Internet voting shares a problem with by-mail voting schemes, which is that there is no mechanism for ensuring that any particular human being only has a single identity. If George Jetson registers his dog Astro and receives mail addressed to Astro, then when voting by mail or via Internet he would have no problem casting one vote for him and one "for Astro". If Mr. Jetson were required to physically show up at the polls and show ID to vote, however, he would have to produce an ID showing that he was Astro Jetson if he wanted to vote on Astro's behalf. Even if he didn't have to show ID, the requirement that he physically visit the polling place twice in order to vote twice would at least raise the possibility of his repeat-voting being noticed by an attentive poll watcher. Internet voting, like mail voting, would remove that safety check.