The LDAP authentication module provides the ability to specify a group based filter for a group which will be admins, and able to see that menu, and the rest.
See the documentation under security.
The superuser_filter and data_profiler_filter are optional. If defined, these configurations allow you to specify LDAP groups that users must belong to in order to have superuser (admin) and data-profiler permissions. If undefined, all users will be superusers and data profilers.
Note that data-profilers can run adhoc queries on any defined connection. They cannot see the admin menu however. You may not want a group of users to be able to do arbitrary SQL or whatever over these, so also set that filter.
Any user can request in their DAG and tasks any variable. It's easy to put those variables in places where they will show up in the logs.
The database provides a way of storing the connection passwords, and variable values in an encrypted way, but that doesn't solve all your problems.