The solution is known as 360-contract. Currently labels profit only from the selling of albums and some of the merchandise, but not from tours, sponsorship deals or ads. An artist on a 360-contract pays the label a cut of everything...
The assumption of the labels is that illegal downloads actually boost the total value of the artist, but diminishes their share (artists make more money on touring but less on the records). The labels have started to push their artists to 360 contracts a while back, but until all the old contracts are up for renegotiation they have no choice but fight the illegal downloading.