Link to a picture of a restored section of the Via Appia.
Is this a good restoration, ie, would it have looked like this? Looks barely wide enough for two chariots.
I'll attempt to answer this question the best I can, but I'd like to say that I am by no means a big-time-hotshot historian; just a dude who loves history, so take my answer with a grain of salt. I am going to take a government centered approach to your question, as it is the best way I can answer it. Though I cannot give you any agreed upon numbers for the amount of people per day/month/year/etc. who used the road.
"The cities and communities normally provided the nearby troops with food, fodder and other requirements"(Peter Kehne)^1. This being the case, and the Appian way itself being in Italy, there would not be a need for supply trains. Thus taking away what could have been a decent chunk of traffic. Emperor wise Suetonius wrote that Augustus "...would change the status of provinces from imperial to senatorial, or contrawise, and paid frequent visits to either sort." Bodyguards would of course accompany him. He also established a system of runners, and later chariots to convey information from provinces to him. Augustus payed with his own money to repave the Flaminian Way all the way to Ariminum. And then asking those who had won a triumph(s) to pay for the re-pavement of other important roads.^2, ^Whether ^the ^Appian ^way ^is ^among ^those ^roads ^Suetonius ^does ^not ^say The Appian Way then likely saw the bulk of its traffic from state functions.^3
I hope that answers your question outside of hard numbers. I cannot go much further than this, which is already an educated guess.
Sources
The Twelve Caesars. Trans. Robert Graves. New York: Penguin Group, 2003. 74-75. Print.^3
The Twelve Caesars. Trans. Robert Graves. New York: Penguin Group, 2003. 62. Print.^2
Erdkamp, Paul, ed. A Companion to the Roman Army. Malden, MA: Blackwell Publishing, 2011. 326. Print.^1
While there is no definitive answer in terms of hard data (i.e. 1,000 chariots daily) - history does tell us a few things.
It was the main thoroughfare connecting Rome to various major cities in the south.
The road was used by the legions to conquer/subdue southern factions.
Other warring parties used the road to attack Latium and Rome. (Specifically Pyrrhus)
Given that these troop movements were large (tens of thousands of men, animals and war machines at a time) and that southern cities like Taranto and Bari were major trading hubs, one could posit that traffic along the Via Appia was quite substantial in the first century AD.
The Via Appia was also nicknamed "the queen of the long roads" which should denote its importance and use.