I have a database that has 128 versions inside. Not all of these versions are used frequently. My question is deleting or reducing the number of versions has an influence on the performance?
Versions are just named states. The size of the state tree impacts performance, even if you have only one version. You'd have to have hundreds of thousands to millions of versions before they would have as significant an impact as a cluttered state tree.
From here http://desktop.arcgis.com/en/arcmap/latest/manage-data/geodatabases/the-compress-operation-and-geodatabases.htm
As a versioned geodatabase is edited over time, delta tables increase in size and the number of states increases. The larger the tables and the more states, the more data ArcGIS must process every time you display or query a version. The greatest impact on performance, therefore, is not the number of versions but the amount of change contained in the delta tables for each version. As a result, versions can have different query response times.
To maintain database performance, the geodatabase administrator must periodically run the Compress command to remove unused data and states. Only the geodatabase administrator can run a compression operation. Compressing performs two key tasks:It removes unreferenced states and their associated delta table rows.It moves entries in the delta tables common to all versions into the base tables, reducing the amount of data the database needs to search through for each version query, thereby improving query performance and system response time.
To maintain database performance, the geodatabase administrator must periodically run the Compress command to remove unused data and states. Only the geodatabase administrator can run a compression operation. Compressing performs two key tasks:
There is also a section on 'archiving' your versions that you don't need but want to keep
Retrieving data ...