First of all, let me say my heart goes out to your sweet son, and also to you! It's so hard to see your child sick all the time, and dealing with it is so hard on you and your entire family (financially, emotionally, etc.).
I had a similar situation with my son (who is 3 now). He was sick ALL the time when he was younger (about 10-13 months). He was in one daycare from 7-13 months (my hubby and I both worked full-time outside the home). He did great for a few months (this daycare was not his original one), but then started getting sick ALL the time!
He ALWAYS had the runny nose, cough, etc. that you describe. And, he had 9 ear infections (two of those were double ones), walking pneumonia and RSV in 3 months! We were all at our wits' end ... we hated seeing him sick all the time, we were spending so much money on the doctor and medicine, and we were both missing work ALL the time to stay home (and keep him out of daycare until he was well).
More than two years ago, we took him for an ENT appt. They tested his hearing (thank goodness he hadn't had hearing loss from all the infections) and for a consultation regarding tubes. The doctor didn't know me, and didn't care to tell me his opinion. He told me that if one of us could stay home with him (instead of sending him to daycare) that his health would improve in 3 weeks or less ... that daycare was his biggest risk factor in getting sick all the time.
We struggled with our decision (and I don't want to say it's been easy financially ... or even emotionally or socially some days), but I decided to quit my full-time job and stay home with him. He was better almost immediately. He's been to the doctor only once since for illness (and he's been at home for more than 2 years now). We have a daughter now, too ... she's almost 18 months old. So, financially, I don't think we'd come out ahead at all having them in daycare full-time if I worked.
That's just my story. I'm not saying that's the only solution ... or that it would work for everyone (child, parent, etc.). I never thought I'd be a stay-at-home Mom (and I'm certainly not saying it's easy ... it's so hard in so many ways). But, I love seeing my son healthy (maybe it was the particular daycare, maybe it's just my son and how his body reacts to even getting colds, maybe he was allergic to something there, etc.) ... and we definitely don't just stay in (so my children are exposed to germs, other children for socialization, etc.). You might want to look at possible reasons he might be getting sick (allergic to something at the daycare site, they might have a weak sick policy and he's catching things from other kids, etc.). If staying home with you kids isn't an option, you might want to look into a different care situation (home-based daycare, a different facility, a nanny, a part-time situation, etc.).
Again, I'm not trying to say that staying at home (or even switching daycares) is your only option ... just sharing my personal story with you (I've been there, and know how hard it is and what you're going through)! Good luck ... I hope regardless of what you do that your son is well (long-term) soon! I wish you and your family the best! ;)