The only way to not have a caret is for the SyntaxEditor.Enabled property to be set to false. What do you need the caret turned off for and still have the control enabled?
I have a case where I want to use SyntaxEditor for an output control. It should be readonly, but I still want the user to be able to select text and copy to clipboard.
In the case where you have a read-only control, standard practice is to display a caret so the user knows where the selection ends. Is it that you don't want the control to get focused when the mouse clicks on it? But then in that situation how would they copy the text if it didn't have focus?