Please follow the below steps:
1. Go to Administration tab -> Service Desk -> Requests/Incidents/Problems -> Status -> Closed
2. In the Closed status detail window click on the tab -> "Request Dependent Attribute Control"
3. Now click on Create New -> for Attribute field select the field name which needs to be required and check the "Required" flag.
4. Click on Save. From now on while closing the ticket the selected attribute will be required.