It's an Asanté card. You can get various versions of their Ethertalk driver and utility suite here. It should be able to detect and install the right drivers for your card. It mentions System 6 support as well.
In hindsight, my Google game is a bit weak, everything I looked for regarding "StNic" or "ST-NIC" pointed to Asanté, but apparently it's because they made a NuBus card dubbed that, after the chipset. There is also a Dayna "StNic" card, but not for the SE. Though the silkscreen font on this card does look like what Dayna used.