Smith also denied laughing at England over the Jonny Bairstow headbutt incident in what has become a fiery build-up to the first day-night Ashes Test, at Adelaide Oval from tomorrow.
Fast bowler Anderson called the Australian team "bullies" in a newspaper column as the tourists fume over the reaction to the Bairstow controversy, following their 10-wicket defeat in Brisbane.
The Australians used the incident, described by Bancroft as "weird" greeting by Bairstow, to get under the wicketkeeper's skin as England collapsed in their second innings.
Smith didn't hold back when asked for his response to England's all-time leading Test wicket-taker.
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"I read the article. I think it's interesting coming from Jimmy calling us bullies and big sledgers," Smith told reporters.
"I think he's one of the biggest sledgers in the game to be perfectly honest with you."
Smith recounted times in his career when he was targeted by Anderson.
"I remember back in 2010 when I first started and wasn't any good, he was pretty happy to get stuck into me then," he said. "Interesting coming from Jimmy."
England skipper Joe Root on Friday said he was "very disappointed" at Smith's reaction and that his team would use it as motivation.
"I certainly wasn't mocking his team. I was laughing at Cameron and the way he delivered the events of what had happened," Smith said.
"He was very dry and different in the way things came across. You guys got a good laugh out of it as much as I did.
"I'm happy to clarify that with Joe Root, no problem there at all."
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Smith, who announced an unchanged team, said despite all the noise he didn't believe the Ashes series would boil over.
"I don't think so. It's an Ashes series, there's always that banter on and off the field," he said.
"I think it's just about playing the game, getting out there and playing cricket and getting on with it."
The Australia skipper said it was up to the umpires and match referee to keep a lid on things on the field.
"From my point of view it's about playing good, hard, aggressive cricket," he said.
"You're only as good as your last game. I out-performed them last game, there's no doubt they'll come out hard again," he said.
"They're good bowlers, they've got good skills. As a collective batting group we'll have to be at our best to score runs against them."
Smith was the point of difference in the Brisbane Test victory with his match-turning unbeaten 141 off a marathon 326 balls.
He said he was prepared to do it all over again in Adelaide.
"If it takes me 300 balls to get a 100, then it'll take me 300 balls.
"That'll also tire the bowlers quite a bit too, so it's a bit of a win-win, if that's the case.