In my Hudson (1.349), when creating a new task, there is an option for Build Triggers , called Trigger Builds Remoteally (for example, from scripts), which, when selected, allows you to enter an authentication token (which has a descriptive text from your question).
I do not know any system configuration that disables this. This does not seem to be a Hudson user rights issue if you have permission to create / edit tasks.
What appears in Build Triggers when creating a new task?
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