The rain itself shouldn't affect the DSL internet connection unless you have some bare copper wires exposed to the rain or there's some connections to those lines exposed to the rain.
Sometimes moisture could also corrode out copper connections on your telephone line which could cause your DSL internet connection to drop when it's raining.
I had the same problem a few years ago with my DSL internet connection dropping so I called my ISP and they sent a technician out to my house to check things over and turns out there was some corroded connections and the telephone box outside on my house was letting some rain water in affecting the DSL connection.
The technician ran a new Telephone wire from the pole to my house and then installed a new telephone box outside on my house and the DSL internet connection still works fine now when it's raining.
So you might have to have your Telephone line checked out.
I would call the ISP and tell them your DSL is dropping when it rains and they can then send someone out to check it out and possibly fix it.
Your DSL shouldn't be going out because of rain.