What would one have to do to head a botanical garden - such as the New York Botanical Garden?"
I assume some kind of botany or plant biology related degree would be required. A few classes to help with running a non-profit business might not hurt, either! I bet if you Google various large city botanical gardens & arboretums, see who heads them up, and Google again to try to figure out their credentials, you could figure it out. Chicago has an amazing botanical garden, and the Minnesota Landscape Arboretum is also great. We like to visit those types of places while on vacation. 
Thanks.
I interned at a public garden a couple of summers ago. A lot of people had horticulture degrees. The director of the whole place had a business background with experience in running nonprofits. However, since my internship, they’ve gotten a new director that has a bachelors and MS in horticulture and an MBA with lots of experience in various levels of leadership in public gardens, arboretums, etc.
I believe if you try to have an internship in the field focused on botanical, you can learn everything, but you must at least have a background on botany. you can also try summer workshops/volunteer work, since summer is the best time to have botanical project and you just need to lookout or wait for possible projects. goodluck!