Global Search displays all records matching the search criteria
When searching solution records from the Global Search application, all the solution records that match the searched text are returned regardless of their statuses - DRAFT or ACTIVE.
This is working as designed.
Resolving the problem
Global Search is created to search all records in the database that match the searched text. To search only the ACTIVE solution records, you can use the Service Request Manager Search application that is located at Go To -> Self Service menu.