To remove just the drop down list, you can only remove actions listed. Through Nuxeo Studio add XML Fragment (in Advanced) and copy that:

  <extension target="org.nuxeo.ecm.platform.actions.ActionService"
    point="actions">
    <action id="non_anonymous_logout" enabled="false"/>
    <action id="login" enabled="false"/>
  </extension>

But if you want also remove the username part you will need to override this resources:

nuxeo.war/incl/nuxeo_header.xhtml

by:

<div xmlns:ui="http://java.sun.com/jsf/facelets">
  <ui:decorate template="/incl/nuxeo_header_template.xhtml">
    <ui:define name="header_user_menu_actions">
    ...Put here what you want that will override the fragment...
    </ui:define>
  </ui:decorate>
</div>

Anyway, be careful, if you remove this part, your user can't logout. So you will need to create your own logout action somewhere…