You stay out of it. As an adoptee, I would be beyond pissed off if someone I was dating decided to interfere in my budding relationship with my bio family. Even my husband of 15 years stays out of it and lets me handle it my own way.
He'll handle this at his own pace, in his own time, based on his own needs and level of comfort. He doesn't need you to decide to take control of his exploration of his past. That's not your place. ESPECIALLY after just 6 months together.
Listening without judgment when he wants to talk about things is absolutely the best support you can offer him, and it sounds like you've been doing a great job with that. KEEP DOING THAT. It's what he needs more than anything.
Keep in mind, his feelings about this are likely things he can't or won't discuss with his adoptive family. Not because he doesn't love them or isn't close to them, but because he's worried that his feelings are somehow a betrayal of the great life they've given him. That's often one of the roughest things for an adoptee who has a good adoptive family - they feel torn between their desire to know their past and their loyalty to the family who loved and raised them.
I don't know who my bio-father is because my bio-mom died before I found my bio-family. I have a ton of siblings and extended family that I've gotten to know, but not a one of them knows the identity of my bio-father - my mother didn't even let anyone know she was pregnant. Not even her best friend had any clue.
To be honest, it's likely that he's not someone I'd have wanted to know anyway. My bio-mother didn't exactly have the best taste in men and, as near as we can tell, between the 8 children there are 5 different fathers. I've come to terms with the fact that I'll never know his identity, but that takes time. I've had many more years to do it than your boyfriend has - I have kids your age LOL
Just be there to listen and to comfort him. And I have one other suggestion. You might look into whether or not there's any way to have his DNA run against the prisoner DNA database. I don't know the status of any of this, which is why you'd need to research it... but if his bio-father was the criminal-type, and has been incarcerated recently, it's likely his DNA is on file somewhere. Now whether or not there's a way to have their DNA compared, I don't know... but you could quietly research that for him and, if it is an option - and ONLY if it is an option - you could then present the info to him as a way of finding the bio-father without him having to worry about hurting his bio-mother's feelings.
Beyond that, your only option is to - OCCASIONALLY - gently encourage him to ask his bio-mother for more information. You might frame it as "her feeling bad temporarily is less important than you going your entire life without this crucial information." Because that's true... she has a responsibility to provide him with any information he needs. That's the Pandora's Box she opened when she decided to establish contact with him.