In addition to the controls that somesoni2 mentioned, you can also hide the Manage Apps menu altogether on a dashboard. I tested this out by creating a new css file at $SPLUNK_HOME/etc/apps/mytestapp/appserver/static/hidemanageapps.css (I had to create the appserver/static directory since it didn't exist by default). Then I added the following text into hidemanageapps.css:
.menu-apps {display: none !important;}
In my dashboard, I added a stylesheet="hidemanageapps.css" into the form element and then restarted Splunk. After that, I was able to go to the page and the manage apps dropdown didn't appear. It's important to make sure that users don't have access to this, in addition to hiding the menu, as we don't want security by obscurity. The approach detailed by somesoni2 will meet that need; in my environment, I was also able to prevent users from getting access to anything undesirable by just inheriting the user role, but you can take whatever approach works the best for you (I have no doubt that there are benefits to the more detailed method).
Update: if you just want to hide the last part of the list (i.e., allow users to choose apps, but not see the manage apps part), you can use the following CSS:
.menu-apps ul {display: none;}
.menu-list {display: block !important;}
