The band would pay if they were putting the cover songs on a CD and profiting from the sales. Another thing for the OP, I believe there is also an exemption somewhere that says if a live band sells their own albums, a license is not required. Just something else to consider.
ASCAP Link: http://www.ascap.com/licensing/licensingfaq.html
The idea is that a business owner is attempting to make his business more attractive to customers by using music. That makes them responsible for the fee.The law says all who participate in, or are responsible for, performances of music are legally responsible. Since it is the business owner who obtains the ultimate benefit from the performance, it is the business owner who obtains the license. Music license fees are one of the many costs of doing business.