Welcome to Just Answer. I'm sorry to hear about this situation. Is this cat eating and drinking ok? How long has this been going on? How old is he/she?
Ok. From that history, I'd be suspicious that you have an upper respiratory viral infection that went from the younger one to the older one. In all reality, some will not kick it as fast as another - similar to a cold with people. A majority of these infections are viral and a few will get secondary bacterial infections. I'll be very honest and say that as long as he's still eating, drinking and not having purulent nasal discharge - I'd hold off on antibiotics. We don't really have any anti-viral treatments. Any time you start to add in antibiotics, you could find the patient gets nauseated, has diarrhea, or is supper stressed by giving the meds. So, I'd be very tempted to say just give him some more time as long as he's still eating and drinking. Let him rest. His body should be able to fight this off pretty well on its own. If at the point antibiotics are needed - I typically will use clavamox (augmentin) dosed at 13 mg / kg every 12 hours. Here's a good link on feline upper respiratory infections.
If there is obvious thick, purulent (literally snot.... pardon the term) discharge - then antibiotics are brought into the picture. Right now, the droplets are just from mild sinus irritation due to the viral infection.