Painting during the Monsoons is a day by day guessing game. You can’t always put off your painting project because it looks like rain. If you did you’d spend a lot of afternoons looking at sunny skies and having nothing to show for a day you should have been working.
In most cases I’ve found continuing with the work plan right up to the first sprinkles is the right move at least 80% of the time. There are a few things that a chance of moisture would cause expensive damage and you wouldn’t want to risk it. Moisture can affect some high tech finishes like urethanes and keep them from hardening or drying by robbing them of the catalyst agents.
Most latex house paints can tolerate a little moisture on the surface during application. Sometimes it’s actually recommended when painting on a hot day. Rain after you’ve applied it can wash it off if it hasn’t dried. Usually, you can just recoat the washed out areas. Once the paint dries on wood or stucco it’s usually fine. Occasionally the metal drip edge paint may crawl or slide if recently painted and it gets rained upon. Once it’s dry is usually is still sticking and a light sanding to smooth it is all the prep you need before you repaint.