You need to write your code in such a way as to keep it running even without Blynk connection.
It has been awhile since I worked with this, and I think is still works (although that was with WiFi-as-Shield)… but as you are using ETH-as-Shield, you might have to do some additional searching/experimenting here about any issues or workarounds when using Blynk.config() with an ehternet connection.