Adobe Reader does not require a password to download from Adobe's web site.
Adobe Reader's installer does not require a password to function.
An installed version of Adobe Reader does not require a password.
Sometimes, some computer Operating Systems are configured such that to install an application or to install an update to an application an "admin" has to sign in.
That, of course, involves a password.
But that has nothing to do with Adobe Reader download, installer or Reader use.
It is completely a thing of the OS configuration.
Be well...