However, when I log on using a subscriber ID I still see the Media option listed below the editor panel and selecting it provides the option to look at any of the Joomla User directories besides their own. Is there anyway to secure this behavior

You can disable the Joomla editor-xtd buttons below the editor, including the Media button, in one of 2 ways:

Per profile - In Editor Profiles -> Features & Layout, set Show Editor-xtd Buttons to No.

Globally for all users - Go to System -> Manage -> Plugins, click on  Filter Options, and select editors-xtd from the Type list. Select the buttons to disable and click the Disable button. You can also edit each button indivudally and set the Access level for it, eg: Super Users.

I'm trying to limit access to only that user's directory. I have the File Directory path set to images/users/$username but this allows the user to back up to the images/users directory. I tried setting the directory filter to +images/users/$username, but this doesn't seem to have helped. Is there any way to lock this down?

Do you have more than 1 profile active? If the profile order is incorrect, the user might be accessing a different profile than the one intended. 

Please send me a login - https://www.joomlacontenteditor.net/contact/site-login

I have the File Directory path set to images/users/$username but this allows the user to back up to the images/users directory.

I assume you mean via the Images & Links tab, with the Intro Image and Full Image fields?

Enabling the File Browser option in the Features & Layout tab of the Editor Profile will convert these fields to the use the JCE File Browser, enforcing the File Directory Path restrictions. I have done this on your site for the Front-End profile.