I think I found a server-side method to detect inside a fragment template whether it is in Edit Mode or View Mode.
It is based on the observation that the URL contains the parameter p_l_mode=edit if the content page is currently being edited. So it basically comes down to get the request from themeDisplay, and then the parameter "p_l_mode" from the request, then checking whether that parameter has the value "edit".Wrapped into a TemplateContextContributor module, it becomes two Component classes:

package my.template.context.contributor;

import com.liferay.portal.kernel.template.TemplateContextContributor;
import java.util.Map;
import javax.servlet.http.HttpServletRequest;
import org.osgi.service.component.annotations.Component;
import org.osgi.service.component.annotations.Reference;
import my.template.objects.PortletHelper;

@Component(
    immediate = true,
    property = "type=" + TemplateContextContributor.TYPE_GLOBAL,
    service = TemplateContextContributor.class
)
public class MyDisplayTemplateContextContributor
	implements TemplateContextContributor {  
  @Override
  public void prepare(Map<string, object> contextObjects, HttpServletRequest httpServletRequest) {
    contextObjects.put("templateHelper", templateHelper);
  }
  @Reference(unbind="-")
  TemplateHelper templateHelper;
}
</string,>

and

package my.template.objects;

import com.liferay.portal.kernel.theme.ThemeDisplay;
import org.osgi.service.component.annotations.Component;
import org.osgi.service.component.annotations.Reference;

@Component(
    immediate = true,
    service = TemplateHelper.class)
public class TemplateHelper {  
  public boolean isEditing(ThemeDisplay themeDisplay) {
    return "edit".equals(themeDisplay.getRequest().getParameter("p_l_mode"));
  }
}

The syntax inside the fragment template then becomes:

[#if templateHelper.isEditing(themeDisplay) ]
    <p>This fragment is disabled during editing</p>
[#else]
    <p>This fragment does something in VIEW mode.</p>
[/#if]

It would work the same way in all FTL templates (not only fragment templates), because we use TYPE_GLOBAL and themeDisplay is already present in all templates.The reason why I would put this into a template context contributor (instead of just using themeDisplay.getRequest().getParameter("p_l_mode") == "edit" inside the FTL code) is that at some point you'll almost certainly need to check the permissions of the user – because any user could theoretically just add "&p_l_mode=edit" to the browser URL, prompting your fragment to possibly show both parts. Such checks would need to go into the isEditing method.
(The issue stemming from the discussion on this page has already been updated with the same text.)