If the heater core leaks it will generally result in the windscreen fogging up when on defrost, a sweet smell in the car (smell of burning antifreeze) and it will usually go from bad to worse quickly, if it busts antifreeze floods the interior quickly. Could the a/c vent tube be blocked or perhaps a leak of water from washing your car,or rain. It is best to look for water leaks when everything is dry. First is the door seal or door, damaged in anyway? I have seen the seat belt having damaged the door seal on many vehicles. Do the doors close flush with body? Ensure the door seal makes good and uniform contact with the door (particularly important if the vehicle has been involved in an accident. You may need to pull back carpets pull back the carpets and sit inside the car (you may need a flashlight) and get a friend with a garden hose to spray water around the door jams to determine if the water is entering the doorway. Then spray water around the windscreen moving around the circumference of the windscreen. Look around the windscreen area and look up from the floor mat area (as best you can) to determine if water is leaking in around the windscreen seal. Is there any rust around the windscreen area or body work? If you have a sun roof there is a drain tube at each corner of the roof to channel away water. A blocked drain may cause water to fall from the sunroof area Water can also come up through the floor pan as you drive through water on the road, if any rubber bung or “seam sealer” has been disturbed or is missing. Remove any debris present in the panel below the windscreen, and ensure any drains for the cowl area there are free and not clogged up or restricted.
Is it water, or coolant? If you have coolant leaking on the floor, then a leaking heater core is nearly a certainty. If it's water, it could be a water leak from a moon roof. Of course, it would have to be raining or you would have to go through a car wash to introduce water that way.
If it requires a heater core, it can be done without removing the dash.
If you plan on undertaking this job yourself, I strongly recommend investing in a shop manual or repair manual so that you can catch all the proper procedures.
this a poor design issue most likely. there is now probably a space btw the windshield and cowl that is allowing rain water to enter. apply gorilla tape over both the top of the cowl windscreen where they meet or rather should meet, and see if the leaking stops.