First, check GL-D for the YE Journals and see if they are in balance.  There was a bug in DBA that caused out of balance YE postings under certain circumstances.  If they YE-Journal is out of balance, UT-A, program GLEREDO (in Evo, GLEREDO.RUN) will recreate corrected YE entries to be posted.
 
If the YE Journal is in balance, check the other journals for all dates, all journal types to see if the overall transaction detail is in balance.  If it is, run UT-K-D to get the balance sheet to match the underlying transactions.  If not, run each journal type and limited date ranges to identify the specific out of balance entry.  UT-A, program EDITGL (EDITGL.RUN in Evo) can be used to make a one sided entry but only after you have confirmed that you in fact need one.

OOPS!!  I spoke too soon!  I noticed that there's a "Net Balance," which does balance.  I'm afraid to say that there has been a long history of this company's Balance Sheet not balancing, long before my time, so I think there's no real way to find out where the problem lies (needle in haystack syndrome).  That's why I was thinking that entering a new Beginning Balance in AM-C might be a good way to re-set it once for all time, if possible.  I will try the other suggestions you made.