I sympathise. You have a very good reason for not allowing your DS to see his father.
IF your DS was younger, I'd say you'd be right to prevent contact. It is a parental responsibility to keep your child safe from harm, and harm is exactly what your DS's father represents ? "criminal, violent and addictive".
However, given your DSs age I just don't think this is a battle you can win. He's old enough to rebel and think he knows better and with some legitimate grounds (it is natural for him to want to know about the other 50% of his inheritance), but he's also too immature to know the way of the world and see just where this could lead him.
I think your best bet is to try very hard to show your DS objective examples of the consequences of getting caught up in this sort of behaviour/family - i.e. drug abuse, criminal record. Maybe you could try some films or books as well as pointing out RL examples. Also work on making clear, short statements about exactly what is wrong with this sort of behaviour (i.e. no vague 'drug taking/drinking is bad' type arguments, but proper reasons like contributing to organised crime and long-term MH problems in the case of drugs, anti-social behaviour, impotence and infertility etc in the case of alcohol, reduced employment/financial potential in the case of a criminal record, etc).
Beyond that, I think you have to allow your DS to explore that side of his family, otherwise he'll simply keep doing it in secret. Hopefully, with your moral code instilled in him and your influence on his analytical/psychology skills, he'll see their behaviour for what it is.
The tricky bit will be doing this and keeping him safe, while also not making him believe that he has to be somehow ashamed of the other 50% of his make-up. Never lose an opportunity to point out what is wrong with your DS's father's behaviour, but also always try to point out positive things about him too (e.g. good at maths, played a musical instrument, clever at fixing cars, good at getting people to like him, etc). Give your DS the impression that the genes he's taken on (the things he's got no choice over) are good, but the bad behaviour is optional and he does have a choice about those.
Good luck. 