David
To support System 7.x and At Ease may have been asking a lot of a Classic with only the initial-release 1MB of RAM (or perhaps 2MB with an expansion card already installed), although it was not an impossibility. When At Ease is open, its icon (an Apple window with launcher buttons in rows) shows in the top right corner of the window, in the menu bar.
Entry of a new password, to replace one that has been forgotten or lost, from the At Ease install diskette will allow an adminstrator access to the Finder and the normal file structure. If the System in use is 6.0.x, however, At Ease will not be the ruling security program.
As has been suggested, System 6 may have had third-party security programs used with it, but the words 'security', 'password', 'file protection' and so on do not occur in the System 6 manual's index. Neither does the disk utility Apple HD SC Setup have any provision for disk-level protection with a password. However, those considerations do not rule out the use of file directory-level third-party utilities, or third-party disk-level utilities either.
Without accurate knowledge about the System and the security utility it is difficult to help you with anything but generalizations.