No, not many at all. There's one right before you get on and one right after you get off, but the only two exits ON the basin bridge are Whiskey Bay (which is highly deserted) and Butte La Rose, which has a beautiful welcome center/rest stop (which I believe is open or has security 24/7 and security cameras) Butte La Rose is closer to Lafayette and the Whiskey Bay exit is between Lafayette & Baton Rouge, if that makes sense.
I've driven that bridge hundreds if not thousands of times for work- if I have to use the bathroom while on it, I will hold it and wait until the rest stop at the Butte La Rose exit (in daylight hours, not sure I'd even stop there at night...probably go to Cecilia/Henderson off the bridge going to Lafayette). No way would I get off at Whiskey Bay even in daylight, just because there's nothing there. Also makes it a better place/obviously popular place for nefarious activity, particularly that connected to missing women.
ETA: Sorry....thought you were asking about exits ON the basin bridge itself.