I don't believe the Internet is making us stupid, but like all technology it does change the way we think. For example, before cars or trains traveling across the country was an expedition, not a trip. Now, however, going from Michigan to Florida isn't a decision about what supplies to bring so you don't die, it's whether to go fast by flying or to save some money and take the scenic route by driving.
I think that people believe the Internet is making us stupid because, like all media, the loudest and most entertaining things are usually the ones that jump out at us. These are usually things that a lot of people might consider "stupid," such as a reality TV shows. People use the Internet like television: to get away from the world. We work, go to school, be responsible, and when we get some time to ourselves we like to get away. I believe it's this getting away that makes people think we're becoming more stupid. They don't consider the time we spend not doing these things, although that time does seem to be getting smaller and smaller so maybe there's something to that line of thought.