I doubt cloud computing fits for their use case. They need constant power, and using "cloud" would be much more expensive than to have their own hardware (and of course even more expensive than using someone's else's computer).
The reason why SETI@home is not popular anymore is because how our hardware adapted. Back in the days the CPUs were constantly operating at the same speed. So if you left your computer on, it was using the same amount of power and generated the same amount of heat. SETI@home took advantage of it, by saying, let's make use of that wasted energy to do something useful.
As computers became faster the heat and power consumption became a bigger issue, mechanisms were added to allow CPU scale its frequency down, and have "micro sleeps" during the time there's nothing to do. OSes similarly adapted. Instead of having an interrupt triggered at regular interval they implemented so called tickless mechanism, when those house keeping operations are fired when needed instead of regular intervals.
All those advancements helped a lot with portable computers (laptops, pads, phones etc) but also made it to the desktops and servers.
Because of those advancements it is no longer "free" to run SETI@home. If you run it, your computer will get warmer, it will turn extra fans too cool down and it will use more power, so people are less motivated to run it. That's what made me stop using SETI@home myself.