Similar to enrolling an individual into a program, you can enroll a family and all its members into a program through the family dashboard.  

Find the Family whose members you want to assign a staff user by using the "Jump to contact..." search bar and clicking the name of the family.

Once on the family's Dashboard, click the button labeled Family Program Enrollment, above the list of family members.

In the window that appears, Select Programs, click the box labeled Select Some Options.

Click on the name of each program in which you want to enroll the family and its members.

To remove a program, click the program name. When you are finished, click Apply to enroll the family and its members in the selected programs. You may also click Cancel to return to the family dashboard.

The family and its members are now enrolled in the selected programs.