Is it a good idea to have a departmental problem database?
I'd consider three key factors.


Are there many faculty members who want to use such a database?  It's probably not worth the effort unless it reaches critical mass.  Taking a survey is a little awkward, since some people may make polite comments that sound more enthusiastic than they really are.  One option would be to ask something relatively demanding.  For example, how many people would be willing to commit to seeding the database by entering all the problems they assign in suitable courses for the next year or two?
Who will serve as system administrator?  Unauthorized access is enough of a cheating risk that it's important to take precautions.  It could be better not to have such a database than to have one managed by an inexperienced system administrator.
Will you be able to work on a scale that doesn't just enable students to cheat by building their own database?  Student groups (such as fraternities) sometimes maintain archives of old exams and even problem sets, nominally for use as practice materials but often in the hope that problems will be repeated.  If you can build a big enough database, then this sort of cheating will be ineffective, but it will work very well for smaller databases.  I think this would be a big problem unless you achieve extraordinary growth, and I'd guess you'd need to work on a far larger scale than a single department.


My take is these are major problems and I'm skeptical about overcoming them.  However, perhaps your circumstances differ from mine.