How about employee number?
SELECT T4.EMP_NUM FROM CLASS AS T1 JOIN enroll AS T2 ON T1.class_code   =   T2.class_code JOIN course AS T3 ON T1.crs_code   =   T3.crs_code JOIN department AS T4 ON T3.dept_code   =   T4.dept_code GROUP BY T3.dept_code ORDER BY count ( * )  DESC LIMIT 1
employee number refers to T4.EMP_NUM; joining tables to correlate class, enrollment, course, and department information; grouping by department to find the one with the highest count of enrollments or associations.