It sounds like a very difficult question. There are so many factors involved, such as laptops, desktops, servers, smart phones, etc. Also, don't forget the routers and the energy to produce all such devices. A pair of researchers from University of California, Berkeley tackled the question and published their results.
Answer: 2% of the whole world energy (somewhere between 107-370GW)!
It might sound a big number but the fact is that it makes up for several more power-intensive activities. For example, the researchers say, attending a meeting physically consumes 100 times more energy than attending it virtually! This is by the way aligned with our mentality in the PI Community: We want to use our infrastructure to make things work more efficiently and get higher value out of our investments.
Where do you think the energy share of the Internet will be in 10 or 20 years from now?