The widespread response comes after Betts posted the footage to social media on Thursday, showing a white car driving by as the youngsters and their friends played basketball.
A man is heard repeatedly shouting a racial slur as the car goes by, before the children leave the court.
"Aboriginal kids deserve to be able to play safely, free from racism and abuse over the fence," Betts wrote on Instagram.
"We are not even safe in our own homes.
"If you know who this is please let me them know that I'm open to having a chat about how much this hurts our kids."
AFL chief executive Andrew Dillon and general manager for inclusion Tanya Hosch said the behaviour has no place in sport or society.
"We must express our strongest condemnation against another example of overt racism, this time targeting children playing sport in their own front yard," the pair said in a statement.
AFL club Carlton, where Betts played more than 300 games, said the incident was vile and it stood in solidarity with the family.
"The footage Eddie and his family were brave enough to release is devastating, confronting and provides just a snapshot of the impact such cowardly behaviour can have," a statement read."Absolutely nobody, especially children, should ever be subjected to what was heard in that video."
Eddie Betts received support from the AFL, clubs and players after posting video of racial abuse. (Con Chronis/AAP PHOTOS)
Carlton coach Michael Voss said the abuse was cowardly.
"I think we're sort of all really tired that it still exists. We'll support him wherever we possibly can.
"Also for the courage he's been able to show to speak out. one thing with Eddie is he always takes a really educational approach.
"He's got amazing humility when it comes to that when the temptation would be to, you know, do something else. Whatever we need to do we'll certainly support where we can."
A Victoria Police spokeswoman said it was investigating after receiving information relating to the incident in Melbourne's southeast on Thursday night.
Betts' post drew thousands of messages of support and horrified reactions with the Western Bulldogs commenting: "Disgraceful. Sending love Eddie."
The AFL Players Association described the incident as "awful" and offered support to Betts, while Victorian Premier Jacinta Allan labelled it "sickening and disgusting".
Betts retired from the AFL in 2021 after 350 games with Carlton and Adelaide.
He was repeatedly subjected to racist abuse online throughout his career and had a banana thrown at him during a match at Adelaide Oval in 2016.
At his retirement announcement, Betts said he was tired of fighting racism and declared the AFL was not a safe environment for Indigenous players.
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