You can join ASCAP as a writer if you've written or co-written music or lyrics for at least one musical work that is available to the public. It could be a song that you perform live or on a CD that you sell at your concerts, a song in a YouTube video, a digital download or sheet music - if it's out there, you're eligible to join.
You can join ASCAP as a publisher if you (or your business, partnership, corporation, etc.) are actively engaged in the music publishing business, and assume the financial risk involved in publishing music. You are also eligible for a publisher membership if you are a writer and have not assigned your publishing rights to someone else. A publisher membership allows you to collect the publisher share of a song's performance royalties.