No, opting to select "Yes, this is my device" will not reduce your authentication security. With the "Yes, this is my device" feature enabled, your device becomes your two-step authentication method and is fully logged for auditing. Having this feature will avoid for a new Duo authentication prompt to be asked every time you login to a Rutgers University service or tool. Some may find this convenient, instead of having to approve the Duo prompt frequently. However, if you log in from a new browser or device, the Duo prompt will appear again. 

There is a minimal chance that your authentication security would be reduced. Someone would need to obtain your NetID & NetID Password AND access to your device in order to compromise your credentials. 

Please note that you will need to have Cookies enabled on your browser for the "Yes, this is my device" feature to work properly.