Qualifications
You need to be a professional musician. Whether that entails a music degree in college, or many, many years of private music lessons and playing in bands and getting out there to jam with people—that's up to you. But there's no way you'll be good enough without some formal music training.
You'll need to be good at music theory and be flexible. Be good at improvising; know the lingo. If a producer says do it in 5/4 time, know what that means. If you need to transpose a piece on the fly, you'd better know how to do that.
You'll have to be an excellent sight reader. Sometimes you'll be given music the night before the practice; often, you'll be seeing music for the first time when you're being paid to play it. Studio time is money, a lot of money, so nobody can afford for you to make mistakes.
It helps to be able to play more than one instrument, which those in the music business call doubling. Pianists often double with the vibraphone and electric keyboard. Drummers can work with any kind of rhythm instruments, from sticks to cymbals, tambourines to maracas, washboards to cowbells. Saxophones usually double with clarinets and flutes, as well as different kinds of saxophones—alto, soprano, tenor. When it's put that way, it sounds more like quintupling. But hey, mo' instruments, mo' money.
Your technical skills will open doors, but your personality and professionalism will be what keeps them open. You need to be able to take criticism without taking things personally. Having an amiable, sunny personality will get you work—way more work than a grumpy perfectionist who's a better musician than you. Nobody likes to work with a kvetch.