From my experience, It is preety hard to clean the dirt or get rid odour once the inner leather start to break. The moist had already get into the leather. Sweat can damage the leather.For me, I will dap a tissue to the leather strap right after a sweaty day or a water splash to my watch during washing my hand. Then let it air dry for a moment. Avoid wear leather strap watch for outdoor activity. I'll only wear my leather strap watch only if I will spent most of my time in an air conditioner area.
Lastly, got a tips from my friend, I will dap a moist cloth to the strap to clean it, and then dry the leather strap using a hair dryer before I store my watch for certain period. It works for me. Next I'm gonna buy some leather wax and use it to prolong the looks of my leather strap.