It seems the default provisioner in the StorageClass does not honour the mountOptions values. I tried to mount a block storage setting dir and file mode to 0777 and they were mounted with root only permissions.
I need to change write permission because my containers run nginx which runs as root www-data.
A workaround is to manually change the directory permission via chmod (either when the container starts with the command + args directives or directly logging in into the container).

Here’s my StorageClass and the PersistentVolumeClaims:

kind: StorageClass
apiVersion: storage.k8s.io/v1
metadata:
  name: storage-rw
mountOptions:
- dir_mode=0777
- file_mode=0777
provisioner: dobs.csi.digitalocean.com
---
kind: PersistentVolumeClaim
apiVersion: v1
metadata:
  name: backend
spec:
  accessModes:
  - ReadWriteOnce
  resources:
    requests:
      storage: 5Ti
  storageClassName: storage-rw
---

Note: in azure aks with a storage account paired to the kubernetes shadow resource group and a similar configuration (changing only StorageClass.provisioner: kubernetes.io/azure-file and PersistentVolumeClaim.spec.accessModes: ReadWriteMany) allows me to create dynamic volumes with the correct file permissions.