Radiator aux cooler: Leaking fittings

I'm going to bet that it is the hose and not the fitting. The greasy accumulation on the fitting hex may have migrated upward on to the hex. Transmission fluid is pretty thin and as it leaks it will pick up dirt. The fresh dirt will soak up the fluid and more dirt will accumulate. Add some wind and a little rain and it could very well look like what you have. I'd replace the hoses any way because of the age and the end looks like it has started to bell from being squeezed by the clamp for a very long time and the heat probably hasn't helped any. You can also make sure the fittings are tight when replacing the hoses.

Make sure the replacement hose is good for transmission fluid as well as the amount of heat it will be handling.