Permission to see Campaigns in Activity search criteria
This is related to this bug 'CiviCampaign: access CiviCampaign' permission missing. Unfortunately at this time, in order to get Campaigns search showing in Search -> Find Activities, you have to grant permission to administer CiviCampaign. 

Depending on your user base (trust and size) and how bad you need this working now, you could give administer CiviCampaign and then change the menu permission for the Campaigns top nav menu item to administer CiviCRM. They would only be able to get to those menu links if they knew the URL directly (unlikely).