If you have a drawing view of a single component, and want Balloons to match a BOM for an Assembly, just right-click on the drawing view of the component and choose "Properties..." from the drop-down. Check the box for "Link balloon text to specified table", and select the correct table from the drop-down. After doing that new drawing views will automatically be linked upon insertion if you're using SW2017 or newer, and as long as there's only one BOM in the Drawing (See SW2017 Enhancement That Didn't Make the "What's New" Document). If you're using an earlier version this will need to be done for each view.
If you have multiple tables in a drawing you might want to name them to avoid confusion when linking drawing views. To name a table, slow double click on it in the tree, or single-click and hit your F2 key (this also works for drawing views, features in Parts and Assemblies, configurations, display states, planes, etc.) Naming drawing views and tables may also be helpful if you save drawings as .PDF's, since these names will carry over as Bookmarks.
[2018-08-31 edit: Beginning with SW 2018, slow double-clicking may or may not work for you for renaming display states (as I understand it, due to fixing a bug that would cause SW to sometimes crash when renaming display states), but single-click and F2 still works.]