Renovation of the neighboring house's exterior wall, who is responsible?

Hello,
the situation is as follows:
To the right of our driveway is the exterior wall of an older house. About 10 years ago, the owner roughly improved the appearance of this wall. In the same case, he apparently rebuilt the wall inside the house and possibly installed a water barrier. Now this wall seems to be damp (the old one at our driveway), the plaster is blistering and falling off extensively. It doesn’t look nice and causes damage. Who is responsible for this wall is clear. But does he have to do anything about it? Apparently, he has no moisture damage due to his own "inner wall." In front of the wall is a gravel bed. How can we persuade him to repair the damage or do we have any say in this?

Regards and

do we have a say here?

No, unless § 921 Baugesetzbuch is applicable (ask a lawyer).

How can one get him to fix the damages

§ 907 Baugesetzbuch 1. Talk. 2. Talk with setting a deadline 3. Letter with deadline in case of own damage occurring (pollution is also damage) 1. Talk and ask for removal 2. Talk and ask for removal within a deadline 3. Letter with deadline 4. Consult a lawyer, lawyer's letter with deadline 5. Document damage, remedy damage, determine damage amount and charge neighbors 6. legal dunning procedure 7. lawsuit 5-7 of course in consultation with the lawyer, who might have other ideas.