First of all, while streaming music from Last.fm is a great way to listen to music you haven't discovered yet, there's no reason to believe that people will lose the urge to collect music.Streaming will never stop downloading
Indeed, the record industry seems to have forgotten the lesson of 70 years' worth of radio: people who hear songs they like often go on to acquire those songs for their personal collections. It's amazing to hear record industry executives deny that this will be the case, especially given that this was the dominant sales strategy for their industry for most of a century. Collecting is easier than it has ever been: you can store more music in less space and organise it more readily than ever before.
People will go on using streaming services, of course. They may even pay for them. But people will also go on downloading. Streaming won't decrease downloading. If streaming is successful - that is, if it succeeds in making music more important to more people - then downloading will increase too. With that increase will come a concomitant increase in Big Content's attacks on the privacy and due process rights of internet users, which, these days, is pretty much everyone.
If you want to solve the "downloading problem" you can't do it by waving your hands and declaring that a totally speculative, historically unprecedented shift in user behaviour - less downloading - will spontaneously arise through the good offices of Last.fm.