The problem is that your drain is clogged. When air is blown across the A/C evaporator coils, the humidity in the air condenses and changes state from vapor to liquid. That liquid has to have someplace to go. It normally drips off of the evaporator coils and drains out to the outside of your vehicle, and when running the A/C you normally would see a puddle forming under the center of the truck when parked if there is enough humidity in the air. If the drain is clogged, i.e. dirt stuck up in it, or dirt and dust is blown in with the air from the A/C / heater fan, that water is just going to fill up the air box and drip out inside the passenger compartment, apparently right where you noticed it in the passenger side front.
Usual fix is to fish some wire in the drain to unclog it, then use some diluted bleach water squirted into the vents to try and deodorize and kill any mold growing in the air box.
If it was a broken heater core, you'd notice a sweet smell from the glycol in the coolant, and a subsequent drop in your coolant level in the car. Also, whenever I've had heater cores go out, they didn't drip out, but put steam inside the passenger compartment through the air vents.
