It seems most likely that this quote, wrongly attributed to Gen. Booth of the Salvation Army, was actually said by the Rev. Rowland Hill, pastor of Surrey Chapel in London, in the mid 1800s. He did not say, "Why Should the Devil Have All the Good Music?" but, "The devil should not have all the best tunes." At that time the quantity and quality of church music was in decline. His message was a "call" for an increase in theologically sound Christian music.
The quote has also been attributed to Whitfield and the Wesley brothers.