According to DMG P.204, Staffs use the spell trigger activation method.
According to DMG P.175, to use a spell trigger item, a character must have that spell on his class spell list. His class level is irrelevant. So, even a 1st-level paladin can use a spell trigger item with Cure Light Wounds spell.
According to DMG P.204, Staff of Healing contain those 4 spells.
*Lesser Restoration (Clr 2, Drd 2)
*Cure Serious Wounds (Brd 3, Clr 2, Drd 3, Healing 2, Pal 3, Rgr 3)
*Remove Blindness/Deafness (Clr 3, Pal 3)
-Remove Disease (Brd 3, Clr 3, Drd 3, Rgr 3)
Thus, a cleric of any level of any level can use all the powers of this staff.
A Druid of any level can use Lessor Restoration, Cure Serious Wounds & Remove Disease.
A Bard of any level can use Cure Serious Wounds & Remove Disease.
A Paladin of any level can use Cure Serious Wounds & Remove Blindness/Deafness
A Ranger of any level can use Cure Serious Wounds & Remove Disease.