About items that cast spells in general:
A magic item, such as certain staffs, may require you to use your own
spellcasting ability when you cast a spell from the item. If you have
more than one spellcasting ability, you choose which one to use with
the item. If you don't have a spellcasting ability-perhaps you're a
rogue with the Use Magic Device feature- your spellcasting ability
modifier is +0 for the item, and your proficiency bonus does apply.
About the drow subrace the PHB says on p.24:
Drow Magic. ... Charisma is your spellcasting ability for these
This could seem to say that the spellcasting ability is valid only for the spells listed there, but the same phrasing is used with classes, for example the wizard:
lntelligence is your spellcasting abilily for your wizard spells, ..
It would be strange if a primary spellcasting class could not use a magic item properly. From this we can say that any source of a spellcasting ability could be used for an item, and only characters unable to cast any spells would use +0.
As such a drow could use Charisma for the DC if no other options are available to him.