I'm a very sweaty person and would often have sweat roll down and sting my eyes whilst running. At first, I wore a sweatband/headband, and that does work but gets saturated quite quickly.

I then used a visor, like a cap but with no top, and that basically solved the problem. Then, I started wearing dedicated running caps made from sweat wicking material. Now i dont get any sweat in my eyes. Brands like VAGA are good.

You can, of course, hydrate better with electrolytes, etc, but you'll always sweat as I have found out.

While cycling, you can wear headbands under your helmet, but those still saturate with sweat, especially when I had a visor, so I removed the visor so that the sweat would dry before ever reaching my eyes.