Your description reads as if you have a MacCon3 NB or a MacCon-i NB-3 card, with Thin (BNC), Thick (AUI) and 10Base-T or UTP (RJ-45) ports. There should be two LEDs: one each for Link (green) and Activity (yellow). The -i variant in each family of cards (viz, NuBus; Centris/Quadra 610, 660AV; IIsi & SE/30; LC PDS; SE) requires both NuBus and LC PDS slots to be present in the Mac.
I deduce that Asanté, as does/did Apple, maintained two series of numbers, for part and inventory, most often with no correspondence between them, which system is undoubtedly more help to the company than to the user. Your card may bear the number 99-00301-02, or more probably 99-00196-01, somewhere, in addition to the 09- number. A MacCon-i NB-10T card should have only AUI amd RJ-45 ports. You need also a floppy disk with Asanté's EtherTalk Installer, which should be installed before Apple's OT 1.1.1 and OT 1.1.2 so that older versions of Apple's networking software from the floppy disk do not overwrite those installed with OT. For best results you do need Asanté's EtherTalk to mediate between the card and AppleTalk. This is in the downloadable v5.6.1 software bundle.
If you are sure of the card's physical installation, confirmed by the Link LED, but get no Activity LED when you attempt traffic between the IIci and another device, you need also to be sure that you have the necessary software, including Apple's extensions (especially Apple Ethernet NB), installed; that the cabling is in place at startup; that you use patch (straight-through) cabling to your 10Base-T hub, or a crossover cable if you are connected directly to another Mac; and that you use a 10Base-T hub between the IIci and a 100Base-T network. This last may not always be essential (different people report different results), but you can be sure that the older Asanté cards otherwise do not always autonegotiate well with every 100Base-T switch or card. I have three IIci Macs with AsantéFAST 10/100Base-T NuBus cards that have no problems in this regard.
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I deduce that Asanté, as does/did Apple, maintained two series of numbers, for part and inventory, most often with no correspondence between them, which system is undoubtedly more help to the company than to the user. Your card may bear the number 99-00301-02, or more probably 99-00196-01, somewhere, in addition to the 09- number. A MacCon-i NB-10T card should have only AUI amd RJ-45 ports. You need also a floppy disk with Asanté's EtherTalk Installer, which should be installed before Apple's OT 1.1.1 and OT 1.1.2 so that older versions of Apple's networking software from the floppy disk do not overwrite those installed with OT. For best results you do need Asanté's EtherTalk to mediate between the card and AppleTalk. This is in the downloadable v5.6.1 software bundle.
If you are sure of the card's physical installation, confirmed by the Link LED, but get no Activity LED when you attempt traffic between the IIci and another device, you need also to be sure that you have the necessary software, including Apple's extensions (especially Apple Ethernet NB), installed; that the cabling is in place at startup; that you use patch (straight-through) cabling to your 10Base-T hub, or a crossover cable if you are connected directly to another Mac; and that you use a 10Base-T hub between the IIci and a 100Base-T network. This last may not always be essential (different people report different results), but you can be sure that the older Asanté cards otherwise do not always autonegotiate well with every 100Base-T switch or card. I have three IIci Macs with AsantéFAST 10/100Base-T NuBus cards that have no problems in this regard.
de