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: 
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.
Take a page from an existing Published Adventure: "Out of the Abyss" 

Out of the Abyss (OotA) has this disease called Grackle-lung. It involves:

* Suffering gradually increasing levels of exhaustion
* Needing to succeed in a Constitution Check to be able to successfully cast any spell with a Verbal component and to take the Dash Action.

It should be noted, however, that the DC for resisting Grackle-lung is pretty low (DC 11 for the first time, and DC10 + exhaustion level after that). And that the disease is a side-effect of breathing in the smoggy smoke of an underground Duergar city with forges constantly alight with Red Dragon fire 24/7, and that the description of the disease says that the phlegm coughed up by victims are black as soot- so a common cold would be considerably weaker. 

Is the common cold really weak enough to not make a difference? If the DC for a disease like Grackle-lung is already pretty low, would that mean the Common Cold is less than 10? Maybe 5? Do you really want to roll a constitution check every time a PC casts a Verbal spell or takes the dash action just to beat the DC of 5? This is for you to decide, I'm just here to tell you there's a disease that does that, but it's pretty exceptional.