Do you currently bike from Brooklyn to Manhattan? I'd be looking at doing that too. I've mainly lived in cities/towns with fairly good biking infrastructures though and am worried about safety
I'll also note that Citibike, New York's bike share program, is a fantastic way to get around lower Manhattan. I use it to commute from East to West Village every day and it's about 10 minutes, and the cost is only around $100 for the year. NYU has several bike stations on campus, one of which is right between Vanderbilt Hall and Furman Hall (so, on top of the library).
I wouldn't rely on Citibike to bike between Brooklyn and Manhattan (the bikes aren't the best for bridges), but in my opinion there is no more convenient way to get around downtown when the weather is nice.