Question: I saw the common cold statted up in the Dragonstar Player's Companion from Fantasy Flight Games. How would the common cold affect player characters in D&D 5e?

I was thinking it would be -1 to str and dex to represent basic aches and pains.

Answer: 
I think that the common cold should be kind of trivial to adventurers whose constitutions are likely several standard deviations above the mean. In the context of D&D diseases, getting a cold seems more like an annoyance than anything else. 

If you want, though, you could use the exhaustion mechanic (PHB291), which is a pre-existing mechanic that's well-suited to this. A single level of exhaustion gives disadvantage on ability checks, which might be consistent with a severe cold.
Given the overall "heroic" tone of modern D&D (doesn't even account for reasonable maiming due to battle wounds, infection from same, etc.), I'd be strongly inclined to leave out piddly stuff like colds, indigestion (unless specific to Plot), corns, bunions, and ingrown toenails, and so forth. Heroes should have heroic afflictions, diseases that will kill you if a quest isn't completed, double-time -- or no ailments at all, short of that.