I take Tal out for a long game of fetch before I leave in the morning because it is the coolest part of the day. The humidity is still very high though. The temperature itself is about 68 so that's not the problem.
I have erred on the side of caution because I know that as humans we don't cool very well in humid conditions. Dog, especially those with dark coast, would be even more at risk. So in the morning I run him for only 15 - 20 minutes tops. Is that too long? Any other precautions?
I always take a gallon jug of water when I exercise Nellie. I fill her bowl and leave it out for her to drink as she wants. When it's hot and humid she can drink almost the entire thing in 1/2 an hour. I'm lucky with her because when she's had enough she will lay down and refuse to retrieve.
I run my kids with a golf cart, sometimes in the hottest part of the day. However, I take them swimming first, jog a ways and then let them drink (sometimes throwing a few pieces of kibble in the water to 'make' them drink). I also have them swim about every 1/2 mile, stopping every 1/4 mile to let them drink and pouring cold water on them and sponging them off in the arm pits and between the hind legs. At the end of the exercise, I hose them off with cool water and try to park my car to catch whatever wind/breeze there is (and in the shade of course). By the time they go back in the car, they really are not panting much at all! My car at work is probably cooler than them being in my house!
I have to be very careful with my blacks. Seamus is a very hot boy by nature and it doesn't take much to wilt him. Flynn and his epilepsy worry me in the heat so evening gentle play is order there. Usually a few kicks of the soccer ball or 5-10 minutes of half-hearted tennisball retrieving after 5:00 PM is okay. When it's still above 90* and 70% humidity in Tulsa even that goes by the wayside. In those conditions we just walk around their yard, at their desired pace, until I see those tongues come out and panting begin.
Paddy (yellow), on the other hand, even at his age, is my stamina dog. Weather conditions in any extreme do not seem to bother him. Hot and humid, icy cold... it's all the same to him (although he prefers that Arctic air to anything!)