I'm looking through the DMG and can't find anything saying what the DC/ Spell modifiers are for the random beneficial properties for artifacts. Does anyone know where that information is? 

The information might not be anywhere, and that does seem to be an oversight.  Perhaps allow the player to use their Charisma if the spell is on the bard, paladin, sorcerer, or warlock spell list.  Wisdom if it is on the cleric, druid, or ranger spell list.  And Intelligence if it is on the wizard spell list.

I might allow the player to choose which stat to use as long as the spell is on the appropriate spell list. 

For example, perhaps the artifact allows the player to cast fireball.  That spell shows up on the Sorcerer and Wizard spell lists.  The character is level 9, so their proficiency bonus is +4.  They have a +1 to intelligence, but a +2 to charisma.  Since charisma is higher, allow them to cast it as a sorcerer spell.  The spell attack bonus would thus be +6 and their spell save DC would be 14.

Alternatively, you as the DM get to choose what stat is used to cast the spell.  From there you can determine the spell attack bonus and the spell save DC.

Note that this may not be an official rule, just one that I would use.