How do I index two table having a one-to-many relation but only the latter is being frequently updated?
The thing about indexes are that they are excellent tools for speeding up queries, but they bring along overhead as well.


If you choose to index every column in a table that is involved in a query, you have chosen to store considerably more data.  This has an effect on the storage required and in the overhead of maintaining all those indexes.  (Or maintaining all columns in a very wide index.)
In past experience, having many indexes, especially several indexes with similar distribution statistics, can make it easier for the server to select a poor index and thus a poor plan. 


My experience shows that indexing just about everything will hurt your performance overall, for the reasons mentioned above. 

Both table A and table B should only contain the indexes that would be of best use for your queries.  

Of course, your first indexing decisions may need to be refined over time as you learn more about your database's behavior.  Do not be afraid to add additional indexes when needed.  And do not be afraid to drop an index that is not being useful.