I have the same problem!
Try installing shades under the eaves to prevent the rain from coming in, or add outdoor curtains that can be closed when it rains.
Ann had some good suggestions for you. Gutters might also help to divert some of the rain coming off your roof from coming in through the screen.
Some install shower curtains on a rod across the each wall that can be slid closed when it rains.
When we were kids we had a screened patio. In the winter my mom put up panels that had clear plastic in them, sort of like a storm window covering.