water leak on passenger side around heater core

Water or engine coolant? If it is coolant the heater core has a leak, if it is water it likely has a problem with the seal around the windshield or a hole in the cowl.

To add to what Bob said, If it's water (and not coolant) it could also be from running the defroster, which uses the AC condenser and produces condensation, which can, if the drain hole for the HVAC plenum box is clogged, leak into the passenger cabin.

Ken, I plumb forgot about a plugged a/c drain. Good call.

Replace the gasket on the plastic shield that covers the heater core, located on passenger side under the glove compartment. There are 5 screws total holding this cover on , 3 visible, and 2 under the shield out of view. Use a good gasket or seal material, like 3/8 dia. x 15' windshield seal.