I went under Users>Enrolled Users in a single course and clicked on the edit icon on the Enrolment Method opposite the user I wanted to suspend and changed Status to Suspend. This was a Manual Enrolment I was editing.
Afterwards the user could nolonger see the course they were suspended from in their My Courses list and (in this case it was a teacher) if they browsed to the course through the All Courses list they could not access it as a teacher but they could attempt to enrol using a different enrolment method such as self enrolment. In this case if they entered the correct enrolment key they would be able to enrol themselves back into the course as a student by using a different enrolment method.
The user could access their other courses just fine. So it worked at a course level but watch out for multiple enrolment methods as it is quite specific to the criteria for which they are suspended.