Physical Danger
The day-in, day-out dangers of bookmaking are similar to those in retail or customer service, or really any other low-risk profession you might consider. You'll be standing a lot, so back issues can creep up over time, and long hours will seem to stretch as you get older. If your workplace allows smoking where the gambling happens, the continued secondhand exposure isn't going to help you stay young (or smell good).
Customers can get unruly sometimes, but ideally there'll be hired security. You can usually spot them by looking for the person on the floor who looks most like a pro-wrestler. They'll be your allies, even if they look like they'd rather tear your arm off and take a bite out of it.
The most dangerous thing you could do as a bookie is join the Dark Side of The Force and enter the world of criminal gambling. This doesn't necessarily mean you'll be working with mobsters with tommy guns and scars and stuff, it just means that your business doesn't follow the law and, more importantly, isn't protected by it. Maybe someone owes you money, maybe you owe someone else money, but the particulars don't matter if the result is you in the hospital with a mysteriously broken leg.