From the DMG at "Activating an Item" (p. 141):
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... your spellcasting ability modifier is +0, and your proficiency bonus does apply.
Thus you calculate your DC as normal: 8 + proficiency bonus + spellcasting ability modifier.
If you don't have a spellcasting ability - and thus have no spellcasting ability modifier - then your spell save DC is just 8 + proficiency bonus + 0. (For the Staff of the Woodlands, which requires attunement by a druid, this is likely only possible for 13th-level Thief rogues, who have the Use Magic Device feature and can thus ignore class requirements on the use of the staff.)