What is the right thing to do for my kids??? PLEASE HELP

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I have a huge dilemma and dont know what the right thing to do is. I have 2 girls who mean the whole world to me. I have raised them without the help or support of their biological father for 9 years now. Just recently the Friend of the Court picked up our case file again. I think the state being so broke now want the money he owes them. He also owes me $60,000 in arrearages. I have not looked for the money, figuring it did me no good, so I directed my efforts towards my children. Anyways, being the FOC is threatening him with a warrant now, he decided to petition the court for supervised visitation. I know that he is only doing this to either get to me, or somehow thinks that his child support will somehow be lowered. I know that he does not have the girls best interest at heart. But he is after all their biological father. He has a huge jail and prison record. For everything from Possession of Cocaine to Fleeing the police. I was absolutely against him messing up the kids I worked so hard to raise right. But am I wrong for fighting his visitations? I think my oldest(11 yrs) is a little curious about him. She said she would want to know what he looks like and sounds like. But I know that as a mother my job is to protect them from all harm and I see him as harmful. He is however the father I picked for them. I just dont know what to do. I am so confused on what the right thing to do is. Should I ask the girls what they want? Are they mature enough at 11 and 9 to know what is best for them, or why I want to fight this or would they resent me later for fighting him on this issue. I want to be the best mother I can be. I just dont know as a good mother if I should allow a supervised visit or fight to keep them from getting hurt? PLEASE HELP ME and TELL ME YOUR THOUGHTS.... I need help with this.