It depends on what sort of 'degree' you're talking about. You can go for a Bachelor of Arts (BA), Bachelor of Religious Education (BRE) degree or simply a Certificate in Christian Studies. You can get a MTh (Master of Theology) or a MDiv (Master of Divinity). I'm not sure what qualifications the various denominations demand, but I think many of the mainstream Christian churches expect that their minister have at least a BA.
You see many whacko TV Evangalists who, I'm fairly certain, do not have a degree in anything except conning people. Then there's the WTS who sends their men to 'elder school' to learn a very slanted view of anything and everything.