The Virginia senator defended Clinton on NBC's "Meet the Press," saying she has acknowledged she erred by using the private email server during her tenure as President Barack Obama's secretary of state.
"She said it was a mistake," Kaine said. "I am not presumptuous enough to start thinking about how I'm going to do things after November. But I know that this is something that she's learned from, and we're going to be real transparent, absolutely."
Kaine also defended the Obama administration over its $400 million payment -- delivered in cash on a plane -- to Iran at the same time the country released US prisoners.
The financial details were announced at the time of the hostage release, though the timing and the fact that it was a cash delivery weren't reported by the administration then.
Sen. Tom Cotton, on CBS' "Face the Nation," said despite the Obama administration's insistence it didn't pay a ransom for American hostages, that's what the rest of the world saw -- opening the door to kidnappings of Americans in the future.
"It doesn't really matter what Barack Obama says. It matters what the ayatollah thinks and what every dictator and terrorist and gangster around the world think," he said.
"And they all clearly believe, in their own words, that this was a ransom payment," Cotton said. "And that means that they are going to take more American hostages, which is exactly what Iran has done since January."
However, Kaine said it was "absolutely not" a ransom payment, noting that the money was part of a settlement of a decades-old dispute over seized Iranian assets.
"There was a legitimate claim against the United States. The United States bargained it down to a fraction. We're making claims in international tribunals, and we're having claims made against us all the time," Kaine said. "We bargain it down to a fraction and paid a portion of the claim. And we got hostages home. Both of those things are appropriate. And I understand why Trump's trying to make something out of it. But there's just no 'there' there."
He turned the issue into an attack on Trump, saying: "The only thing new this week, the only thing new, is Donald Trump's claiming to have seen a nonexistent video of cash being delivered to the Iranians. That's the only thing new that's happened this week. He persisted in making that claim again and again until he finally got shamed into acknowledging that there was no such video."
Trump had claimed to see such a video several times last week, but admitted in a tweet Friday morning that it was actually footage of the plane carrying US prisoners arriving in Geneva, Switzerland.
"The plane I saw on television was the hostage plane in Geneva, Switzerland, not the plane carrying $400 million in cash going to Iran!" he tweeted.
Kaine was also pressed on a major split on the Democratic ticket: While both Clinton and Kaine have urged Congress to approve a new authorization of the use of military force against ISIS, Clinton has said it's not legally necessary, while Kaine has asserted it is.
"I have very grave doubts about whether the legal authorities currently in place allow us to wage an offensive war against ISIL," Kaine said.
"But Secretary Clinton and I get to exactly the same spot, in that Hillary has said, now this probably goes back six or eight months now, that Congress should finally own up to its responsibility that is the most solemn responsibility in Article One of the Constitution, that authorizes military action," he said. "Our troops deserve it. Our allies need to know it. And our adversary needs to know it."