The easiest thing to do is set up a batch job on your server, kicked off by the scheduler, which attaches the disk on a workstation, copies the files or directories it wants to a directory on the server, then disconnects the workstation, and moves on to the next. The NET USE command maps the drive, NET USE /DELETE releases it.
PROS - you do not need to keep drives mapped all of the time. Users do not need to copy files.
CONS - users need to leave computers on during the critical period. Server needs to know which areas/files to copy. Server needs rights to user systems.
Alternate solution:
Each user has a network folder, which is automatically backed up. User is responsible for putting important stuff in the folder.
PROS - simple, no rights needed, user can turn off machine.
CONS - the obvious -- user does not remember to copy files.
Other backup software, most of it expensive, may have a client agent on each workstation to trigger a backup (Legato, Archive and others). The user stations still need to be turned on.
Good luck.