Distrubution1 Database Offline - I have no clue what this is
Offline is offline; the database can't be "doing anything" unless there is application code somewhere that changes it from offline to online and then back again.

As to whether you can drop it, well, we have no idea. It may be cruft, but it may be important. (Think about me asking if this cable I found in my basement goes to anything important - I wasn't actively using it, obviously, but I may need to charge some device tomorrow that needs it.)

To get a better idea of what this database is, you can try to get some clues or trigger your memory in various ways:


Take a look at revealing properties, such as create date / owner.
Bring the database online temporarily, and inspect the names of tables and the data inside them. If you are afraid to bring it online long enough to inspect it, you can bring it online just long enough to set it to single_user and back it up, then take it back offline, then restore a copy as a different name, and inspect the copy.
Review the default trace (see some examples here) to see if there are any significant events (but the default trace rolls over, so it may not go back far enough).