The 24-year-old has been embroiled in a number of off-field incidents in the past, most recently when being found guilty of assaulting a police officer last September.
Aurier, who also aimed a homophobic slur at Laurent Blanc during the pair's time together at Paris Saint-Germain, is now attempting to get his career firmly back on track in North London following a summer switch.
Drogba wants to see the full-back stamp out his problems, telling The Mirror: "Serge Aurier maybe has to make those mistakes to really understand and realise what he represents. He's a good person, he's loyal, he wants to succeed.
"His problem is away from the pitch – he's always given everything on the pitch. But you know that when there are kids who call your name and ask for autographs, you have to be careful with all your actions and gestures."
Toure, another of his former international teammates, added: "It's a lack of maturity, quite simply. I think he must feel very bad about it. But he got carried away and didn't realise the consequences.
"Serge is a really good guy, ask his teammates. He's young. We've all been like that. We've all been through that stage when we're young, playing the hard guy."
Aurier has featured four times for Spurs since joining, picking up a red card against West Ham United during that time.