There are no "requirements" for BASE jumping. Absolutely nothing and no one is stopping you from purchasing a rig and hucking yourself off a bridge, cliff, etc.
However, the way 99.9% of all participants get into BASE is through sky diving. You need to learn what falling feels like, how to handle a chute, how fast you fall, and a wholeeeeee lot more. To get into skydiving you'll likely start out with a jump or two strapped to another dude. If he gets an erection, he's gay. If you get an erection, you're gay. However, this does absolutely nothing to qualify you to jump on your own.
To learn/qualify to jump on your own, you'll need to take a course. This will cost a fairly significant amount of money. Afterward you'll be able to get on a plane and jump on your own. Each trip up (each jump) will run you only around $25. This assumes you have your own rig (parachute). It's strongly recommended that you have several hundred jumps under your belt before you attempt your first BASE jump.
There's a bridge in Utah that is super friendly/easy to jump and many folks do their first BASE jump there. An important consideration is that you will need a BASE-specific rig. The rig you use for skydiving will not work. The natural progression in the sport, if you wish to get into wingsuit flying, is to wingsuit from a plane a significant amount before attempting it on a BASE jump. You'll also want a really significant number of BASE jumps under your belt before you consider a wingsuit.