Hi,
The Epson ET-3850 is listed in About AirPrint - Apple Support as a model supporting Apple's AirPrint technology. This means that the printer should work with macOS without any additional drivers, provided that the printer and the Mac computer are connected to the same network (Wi-Fi router).
Is your macOS version 10.11 (El Capitan) as indicated by the above equipment line? If not, please post the correct macOS version. AirPrint should work with macOS 10.11 El Capitan as well; special software from the printer manufacturer may or may not.
Since you can print from an iPhone, would it be correct to assume that the printer is properly connected to a Wi-Fi router? That is, not some kind of Wi-Fi direct method (iPhone to printer, or Mac to printer, without a router).
AirPrint works with wireless (Wi-Fi) or wired (Ethernet) network connections, so you could choose to wire the printer by Ethernet cable to one of the LAN ports of the router instead of using Wi-Fi. Ethernet may be more stable and could eliminate some of the issues related to a wireless connection. You could try Ethernet, at least for a comparative test.
Also, in case of difficulties with AirPrint device detection (if all settings are OK), you may want to try the following procedure in the described order (do not forget the router):
Power off the Wi-Fi router, the printer, any Macs, iPhones, iPads, and all other computers and devices. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two.
Power on all Macs, iPhones and IPads. Finally, power on all other computers and devices.
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)