Reed switches are normally SPST so how do you arrange the magnet and reed so that the reed is open when the door is closed?
An alternative solution would be to use a SPDT reed (if you can buy one) so that the reed is "operated" when the door is closed and so the N/O contact is open. Thus when the door is opened, the reed "releases" and thus closes the buzzer circuit.
Another possible problem is that the reed may "stick" due to residual magnetism and therefore won't "release" when the door opens. If this proves to be a problem, another magnet (fixed in the door jam near the reed) could arranged to give a small amount of reverse flux when the door is open.