Bao-Dur isn't hard to convert, probably the easiest after Mira, but he can be a bit tricky since his stats aren't the greatest so people don't take him along.
Simply put, you simply have to have him along at certain points in the game where you can get influence with him. You don't get influence with the guy for talking with him. I think you need like only two or three influence raises with him then he's yours. It's easiest to do this on Dantoonie as a LS or neutral character. Make sure he's with you when you get the quest to find the atmospheric sensors in the Kinrath Cave (a.k.a. "Crystal Cave"). He'll volunteer to look at them and when he finds the spy devices, be sure to tell him he did a good job (if your skills are high enough you can do it yourself and he'll be impressed anyway).
Once you're done doing whatever stuff you went to the cave to do, step back outside and go back and talk to the militia fellow who gave you the quest. Be sure not to have anyone else in your party that might "sponge" the influence conversation. It doesn't always happen, but a lot of influence oppurtunities are only triggered for one NPC. If you want to be sure, remove the third person for your party for your conversation.
When you speak with the Militia fellow, bring up the fact you found the spy sensors and make the guy come clean. When he offers you 5,000 for them, tell him they need the money and only take 3,000 (I think it's 3,000). When the conversation ends, Bao-Dur will be impressed with you again.
Then start talking with Bao-Dur over and over again. Once you get the line "I think you're having an effect on me, General" you have him. You might need another influence raise, but I don't think so. If you do, places where you can get influence with him (off of the top of my head) are helping the Mandalorian on Dxun and helping with the airspeeder on Nar Shaada.