A lot of it depends on the natural history of the species. Usually, the animals (expectant mothers) will let you know what they need ... either consistency with their normal routine, or that they just want to be left alone.
With some herd species (caprids - sheep and goats - jump first to mind), the routine stays pretty much the same all year round. Babies are expected in the spring, but there are no real preparations since the babies are precocial and can go from day one. They are usually born in the exhibit with the group.
Solitary animals and/or those with altricial young require more preparation. Since the babies can't fend for themselves when born, the mother will often prepare a den in which to give birth. In order to facilitate this, the female is often separated from her cagemates before the birth.
Monitoring is done on a case-by-case basis. Remote cameras (but not linked to the web) are used for the high-profile animals like the black-footed ferrets and Vancouver Island marmots where disturbance could be very bad. For others, keepers simply check in on the animals regularly. Most babies tend to be born at night, so often on morning rounds there will be a new addition. Assistance is almost never required ... the animals are left to get through the process on their own (without interference).
