When you become Wanted, you get a bounty placed on your head, and lose the ability to Give Up in a fight. The bounty is equal to 10,000 G times the week number. You also cannot land on any towns, castles, temples, shops or Dokapon Castle.
In Dokapon DX, the Sheriff is notified, and will try to hunt down the Wanted player.
In Dokapon Kingdom, the Accessory "Criminal Studs" will give a temporary raise in stats if the Wanted status is active. The amount of days Wanted can also be reduced if the "Nice Guy Badge" Accessory is equipped. The Ninja end of week bonus involves Attacking/Robbing a Shop. Even if you fail to rob something, you will still get the bonus.