It's a pain and I"ve been told the only way to fix the leak is to replace the cowl. This is a massive job and involves drilling out hundreds of spot welds. One enthusiast counted them one day I remember being shocked at the number of welds that need to be drilled.
Anyway, I have seen a specially made clear plastic cover that is fixed on the external cowl vents that seals the cowl area and stops the leaks! It's only a bandaid fix but it stops the leaks if your happy to see it on the vehicle . Not sure who supplies these plastic cowl seals but I'm sure someone can point you in the right direction. Alternatively , if you are handy, it's just a piece of clear Perspex with some foam around the outer rim fixed using a T clamp & a small nut & bolt.
I"ve also heard of a plumbing sealer type substance used to repair leaky pipes can be poured into the cowl from the outside vent but this seals small holes ONLY ! I haven't tried this because I ended up selling my 65 stang.
If it leaks it ha a hole in it. More than likely perforated with rust. Reach up under the dash and feel the bottom of it and check things out. I have seen windshields leak and look like a cowl leak, but then again a windshield leak can be an indicator of rust at the windshield pinch weld at the cowl flange.
they make a repair for it .it is a plastic ring that extends inside the hole and silicones to seal you have to remove the heater box or the air vent, what ever side it is that is leaking i would do both sides. any parts vender should have them,National parts,Mustangs Unlimited. i have used them before they work
I did the plastic clear cover 20 years ago in high school for my 67 and it worked great. Only had to install it when it was raining or calling for it. I lost the cover a long time ago, but the car sat in the garage for years... I got her out 7 years ago just to get her running before I tried to do some updates to her. I installed the plastic cowl hats... They didn't work for me as the cowl was too far gone. I reached up and pulled the hat out of the car on the drivers side the passenger side was pretty decent only small pin holes. I finally broke down and have been doing the full cowl replacement, well just the 2 pat pieces and the top. My stock cowl top was shot at the corners. It looked like from the factory they filled the drain on the drivers side with seam sealer. Now don't get me wrong it is a pain, but I don't think that it was any harder then doing the front core support and front inner fenders. Maybe I just went into it thinking from all I read it was gonna be a lot harder than it was.
But in my opinion it is worth it to just do it right and not band aid it. That is coming from someone that did the band aid fix for many years.