If you are on version 7.0 and above, We are currently evaluating this issue and will update this article when we have more information.

This can be accomplished by setting up a role, task group and security assignment, and assigning this role to the form's Targeting and Security tab. When you assign a role to the Targeting and Security tab of the form, the submitted form data will inherit the form's Targeting and Security. This will allow only users who have this role to see the submitted form data. 
Note: this change is not retroactive: editing the targeting and security of a form will not change the view of any previous form submissions.

Create a role:
  1. Navigate to Users & Security>Roles
  2. Click New Role
  3. Enter a Role name
  4. Click Save
Create a task group:
  1. Navigate to Users & Security > Task Groups
  2. Click New task group.
  3. In the window that pops up, select Site Task Group and click Okay
  4. Enter a Task Group Name
  5. Under Form tasks, mark the box for View rights for Data tab only
  6. Click the green button New Security Assigment
  7. Select the radio button for Site Tasks
  8. Choose the newly created role
  9. Choose the newly created task group
  10. Click Save

Add this new role to your Targeting and Security of the form:
  1. Navigate to Site Explorer>Forms
  2. Locate the form in question and click the Pencil icon to edit it
  3. Select the Targeting and Security tab
  4. Unmark the view for Everyone
  5. Click Add Users & roles
  6. Select the desired role from the left and click Add, then mark the desired boxes under assign permissions
  7. Click OK
  8. Click Save
For more information in regards to setting Targeting and Security for the Form Display part and to set who can view and fill out the form