IMO E is the correct choice
NCC
Despite public health campaigns with the aim of achieving a reduction of unnecessary prescriptions for antibiotics, the drugs continue to be prescribed at startlingly high rates in outpatient settings such as clinics and physician offices in the United States.
(A) Despite public health campaigns
with the aim of achieving reduction of unnecessary prescriptions for antibiotics, --> Redundancy
(B) Despite the
fact that public health campaigns were aimed at reducing unnecessary prescriptions for antibiotics, --> Also were is the wrong usage
(C) Despite public health campaigns,
which were aimed at reducing unnecessary prescriptions for antibiotics, --> The red becomes non-essential clause as per the framing
(D)
Despite of the public health campaigns aimed at reducing unnecessary prescriptions for antibiotics, --> Unidiomatic structure
(E) Despite public health campaigns aimed at reducing unnecessary prescriptions for antibiotics --> Correct