What is the ruling of Islamic law on paying a sum of money to a brother who is disobedient to his mother and addicted to drugs, and unemployed, in order to prevent him from being imprisoned, knowing that he has five children?

What has been mentioned about your brother regarding his disobedience (to parents) and drug use are major sins that incur Allah's wrath, lead to unemployment, abandonment of work, and accumulation of debts. He must be advised and warned about Allah, reminded of repentance, and you should make abundant supplications for his guidance, especially the mother's supplication, which is answered. His transgressions do not prevent you from showing kindness to him, without aiding him in wrongdoing. You are not obligated by Sharia (definition) to pay off his debts, but helping him with that, or part of it, is a good deed that may be a cause for his guidance and acceptance of advice.