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.
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.