You need to add the httpNodeAuth (not the httpAdminAuth as this is for controlling access to the Node-RED editor and can done with the environment variables discussed in the other answer.) to the app/bluemix-settings.js file.
Something like this:
...
httpStatic: path.join(__dirname,"public"),
httpNodeAuth: {user:"user",pass:"$2a$08$zZWtXTja0fB1pzD4sHCMyOCMYz2Z6dNbM6tl8sJogENOMcxWV9DN."},
functionGlobalContext: { },
...
Details of how to generate the pass can be found here
There are a number of ways you can edit the file, some of which include linking the Node-RED deployment to a git repository or downloading the whole app, editing the file and pushing it back to Bluemix (when you first deploy Node-RED from the starter pack it gives you instructions on how to download the source to make changes and then push them back. You can get to these instructions by clicking on the "Getting started" link in your Node-RED Bluemix console page).
But the quickest/simplest/dirtiest way is probably to just SSH into the instance and change the file with something like vi. Details on how to ssh to an app instance can be found here. But the following should work:
cf ssh [app name]
Once you have edited the file you will need to tell bluemix to restart the app. You can do this from the web console or with the cf command line tool.
(The changes made by this method will not survive if the app is restaged, or bluemix decides to move your instance to another machine internally because it will rebuild the app from the pushed sources. The permanent solution is to download the source, edit and push back)
