In the most extreme case, in which one parent has been convicted of first degree murder of the other parent, the law specifically prohibits visitation with the children until they are of a suitable age to assent.
Similarly, but to a less serious degree, in making custody and visitation determinations the court will consider crimes that would cause one to question the fitness of a parent. These types of crimes would obviously include any violent crime convictions which could call into question whether the children would be in danger around a parent who has shown themselves to resort to violence when faced with conflict. In addition, drug and alcohol abuse offenses would call into question a parent's ability to care for their child without supervision.
Other crimes that might seem unrelated, such as theft or prostitution, may not cause a Judge to question the parent's ability to care for the child, but rather their suitability as a role model. These types of convictions might be a reason to limit time with the child, but may not require supervision (assuming that this criminal behavior is in the parent's past).
Judges have access to the criminal record of parents, and when issues are raised like those discussed above, the court will often request the records of both parents before making a determination. This is also true in any 209A Restraining Order cases.
It is also important to note that the Criminal Offender Record Information (CORI) system is changing in 2012. All of the changes are listed here. Some of the changes that could affect your case, include a new procedure for having an inaccurate record amended, and new permissions for access to sealed CORI by court order in domestic abuse/child custody actions and where a person’s safety is at stake.