--don't let an error rollback your explicit transaction
set xact_abort off 

set nocount on

create table #t (database_id int)

declare @dbid int

continueCursor:

select @dbid = isNull(@dbid, -1)
begin try
       begin tran

              declare crs cursor for
              select database_id from sys.databases where database_id > @dbid
              open crs
              fetch next from crs into @dbid
              while @@fetch_status = 0
              begin
                     --Do some work
                     insert #t values (@dbid)

                     --Raise a test error for one of the values
                     if @dbid = 4 raiserror('test error', 17, 1)

                     fetch next from crs into @dbid
              end
              close crs
              deallocate crs
       commit tran

end try
begin catch
       select error_message() [Error]

       if cursor_status('global', 'crs') > -1
              close crs
       if cursor_status('global', 'crs') = -1
              deallocate crs

       goto continueCursor
end catch

select * from #t
drop table #t