I would check all of your extension cords and connections. If it is possible, and if they are not plugged into a gfi outlet, plug them into a gfi. If you have a connection that shorts to ground when it is wet outside it should trip it. If it does trip you need to get some new cords that are rated for outdoor use and try that. It sounds like water is shorting something out and that is causing your problems.
It is possible the water from the rain is seeping between the plugs and tripping the gfci outlet that you are plugged into. Try allowing them to dry then tape sandwich bags around the plug connections and reset the gfci.
You are not using outdoor lights which are specifically designed and made to run outside and even in rain. There is a conductivity problem. To know about the details, how to make the lights work in rain, visit this web page