Luckily I don't have a thumb sucker and my 20 month old gave up the paci all on his own around 10 months. But if he HAD decided to pick up this gross habit, I would've stopped it before it became an addiction. It grosses me out to see kids 3,4,5 and older walking around in stores sucking away at their thumb. I always wonder how in the hell those parents aren't embarassed OR at least only allowing it at home.
Nowadays everyone's trying to debunk the FACTs about the damage caused to kids teeth by thumb sucking, saying it won't harm them til after age 6. Google image search "thumb sucking" and SEE the bad teeth for yourselves! I don't know why but seeing kids, even toddlers, with their entire thumb down their throat grosses me out. It's not cute, its annoying and looks weird. Experts say don't punish your kid because its a self soothing thing, but I honestly believe if my kid were sucking their thumb day in and day out, and nothing was working to make them stop, I'd use some sort of punishment.
Just like for any other bad habit. Picking your fingernails, coughing without covering your mouth, burping without saying excuse me, feet on the table etc. when you say "don't do that" over and over and your kids are clearly able to grasp what you're telling them NOT to do and they don't LISTEN, you take a toy away. You make an earlier bedtime. You say no friends over. No dessert. Whatever! Just make them stop! You're the parent!