Can I stop her from being around my child with her having a drug background?

Your question cannot be answered in this forum because more information is needed.  Lawyers.com offers general information and not legal advice.   As a general rule, when courts are called upon to deal with matters of custody and visitation, they are concerned about the overall best interest of the child.  There is a presumption that the best interest of a child is served by healthy contact with both parents.  If you file a petition, seeking to modify or impose restrictions on the current visitation, the burden is on you to show why your requested action is in the best interest of the child.  The Court will refer to the best interest statute and make a decision based upon the facts in your case, as applied to exist law.  As a practical matter, what if any steps have you take to address this issue with the father?  Have you raised your concerns with him?  If so, what has been his response?  What information do you have showing that the new girlfriend is around your daughter? Is she high around the child? Does she have a drug problem?  I suggest that you follow up with the father, before taking legal action.  If you are unable to resolve this matter with him, I recommend consulting with an experienced family lawyer in your area to schedule an office consultation.  Most lawyers charge by the hour for office meetings, therefore, you should work all payment details prior to setting up a meeting to avoid any misunderstandings.  Best of luck~