Many indoor places, including aged classrooms and offices, prisons, homeless shelters, etc., are poorly ventilated but resource-limited to afford expensive ventilation upgrade or commercial air purification systems, raising concerns on the safety of opening activities in these places in the era of COVID-19 pandemic. To address this challenge, using computational fluid dynamics, we conducted a systematic investigation of airborne transmission in a classroom equipped with a single horizontal unit ventilator (HUV) and evaluate the performance of low-cost box fan air cleaner for risk mitigation. Our study shows that placing box fan air cleaners in the classroom results in a substantial reduction of airborne transmission risk across the entire space. The air cleaner can achieve optimal performance when placed near the asymptomatic patient. However, without knowing the location of the patient, the performance of the cleaner is optimal near the HUV with the air flowing downwards. In addition, we find that it is more efficient in reducing aerosol concentration and spread in the classroom by adding air cleaners in comparison with raising the flow rate of HUV alone. The number and placement of air cleaners need to be adjusted to maintain its efficacy for larger classrooms and to account for the thermal gradient associated with human thermal plume and hot ventilation air during cold seasons. Overall, our study shows that box fan air cleaners can serve as an effective low-cost alternative for mitigating airborne transmission risks in poorly ventilated spaces.