You need to use outer joins in case the student doesn't have courses in one of the categories:
Select nvl(sum(C.Credit_Hours),0) As CoreCreditHours,
nvl(sum(C1.Credit_Hours),0) As ElectiveCreditHours
FROM COURSE_TAKEN
LEFT JOIN COURSE C ON
C.CRN = COURSE_TAKEN.CRN AND C.CRN IN(
SELECT MINOR_CORE_REQUIREMENTS.CRN
FROM MINOR_CORE_REQUIREMENTS
WHERE MINOR_CORE_REQUIREMENTS.Minor_Name = 'Computer Science')
LEFT JOIN COURSE C1 ON
C1.CRN = COURSE_TAKEN.CRN AND C1.CRN IN (
SELECT MINOR_ELECTIVE_REQUIREMENTS.CRN
FROM MINOR_ELECTIVE_REQUIREMENTS
WHERE MINOR_ELECTIVE_REQUIREMENTS.Minor_Name = 'Computer Science')
WHERE COURSE_TAKEN.H_Number = 1;