Can my damaged Braun electric toothbrush with Duracell batteries explode?
No. Don't worry. The worst thing that can happen is your appliance stops working. 

Maybe you'll find someone that will say that there's a little chance of it explode, etc. But the chance of explosion is the same chance of one asteroid falling over your house, and if it happens the toothbrush is the lesser thing that you could worry. ;-)

With only two cells and with the kind of chemicals those cells use, there's no risk of explosion, neither of electric shock.

Don't be afraid: just put some glue or tape to close this gap and keep the water to get inside (it will be very ugly) and keep using it, if you want. It will not harm you, period.