Is it unprofessional or a bad idea to use my department mailing address for personal mail? I will start a PhD in another city next month, and I need to update my address with the USPS, my bank, and so on. Since I'll be living in graduate housing and expect to move a few times during the program—say I take a summer internship, and graduate housing assigns me to a new room the next academic year—, I would like to make my permanent address my department address. Is this unprofessional or a bad idea? I would be receiving credit card statements, junk mail, and everything in between at my department address. Alternatively, I could just keep changing my address with the USPS every time I move.
Check with the department/school first.  

In general for use of any work address:
For something like receiving a single package, it should be fine. 
For something like receiving regular mail on a daily basis, no you should not do this.  Besides the issues of policy, there is an increased labor of the recipient to deliver your personal mail.  There may be delays.  And, normally these boxes are in a partially-public area and subject to theft.  

I've been in situations like this, so I got a PO box once and a private mailbox company another time.