Thats what I do. I wear one of the 1 $ face masks you can get at lowes or home depot or walmart and wear gloves. Bf is usually pretty good about keeping it clean but when im on a cleaning binge I just do it myself.
Wearing gloves is fine. Also, don't inhale. Put something over your mouth and nose while doing it or just put your shirt over your nose. I cleaned my cats litter all the time and baby came out fine!