As hard as that might be for you, yes you should. Provided she's a decent person and won't cause them any harm.
and pays child support each week
You have the right as their mother to set boundaries. She does not have the right to discipline your kids or play parent in anyway. Setting boundaries is vital. These are YOUR children and you are their only mother, they must be your first priority and you DO have the right to say that his time with them should be without her if you have reason to feel that way.
If he's serious about her than yes. It eases the situation overall for your children to know who their father is seeing instead of keeping them in the dark--even if it is difficult for you
Have you met her? I would want to meet her first, then I would allow it, I would also think you would do the same for him.
Thats a reflection on him, not you, he shouldnt be letting antone around your kids unless he is sure he is going to let them stick around, I see where you are coming from now.