I would let things run the natural course.
So ask yourself the following question: is someone going to miss the Staff? The person who put it there, other people who were trying to break the vault?
If the answer is yes, then you have someone searching for the PCs, which may want to take it back from them by good or evil means. Someone with a crazy vault and a Staff of the Magi is probably quite a powerful/rich person, so I would do indirect stuff first: bounties, low level henchmen... This can prompt the players to search help from other groups or if they don't get the cue, make other groups contact them offering help. Escalate things slowly though, it can take sometimes a lot for players to realize they are punching above their weight.
If the answer is not, then basically life is business as usual for the PCs, no one really knows they have the staff. And this should continue like that unless they do something to let it be known. If they use the staff, but they are no witness, let them have their fun. The staff works with charges, so you have some leeway there to handle it.
Basically, continue things as normal, don't act differently because they have a Staff of the Magi (don't change monster tactics, encounter layouts...). If people come after them, make it clear it is because they stole something other people want, not because it's an item they shouldn't have.
And also, accept their decisions. If you make clear having the Staff is dangerous and that bad people are hunting them, and in the end they decide to hold it, well, their call. Imagine they surprise you again, and they manage to survive and hold on the staff, it's something they are going to remember forever.