If you do a Join between the Feature Class and the Attachement table to identify which features have attachments or not the resultant layer will not expose the attachments when using the Identify tool. This means we can symbolize them as needed but we loose the functionality of retrieving the one or many attachments since a join is a one-to-one and attachments are a one-to-many relationship.
Xander has a good process to identify those features that have attachments and make note of it in a new field in the feature class but this is a job that needs to be run after every edit or built into a users workflow.
An on the fly way to accomplish the symbology we are after without extra fields and jobs to calculate those fields is to leverage the layer's definition query. ESRI's documentation refers to this as a Subquery when you build a definition query for a layer that has to do with data from another feature class or table. Try using this syntax to complete the SQL statement in the Definition Query property of your layer to filter its records based on the presence or absence of attachments. In this example Global ID in the Feature Class is used as the foreign key in the Attachment Table.
Feature with Attachments = GLOBALID IN (SELECT REL_GLOBALID FROM <your attachment table>)
Feature without Attachments = GLOBALID NOT IN (SELECT REL_GLOBALID FROM <your attachment table>)