Then, my guess is that for normal, healthy ink, it should suffice with good hygiene. Nothing fancy unless one lives in a nasty location. I still keep bottles I opened 30 years ago containing clean ink. You can guess that some inks I do not use much.
What I do: I just keep them in their box in a drawer (not exposed to light), open only to fill the pen (I always use converter or piston pens), introduce the pen tip, fill, take pen out, put pen on a rag (to avoid staining the table) and close the bottle and put it back to storage. Then use the rag to wipe the pen section of ink to avoid staining my fingers (which I invariably stain anyway).
I follow that order for I know my careless self and am always terrified I will hit the bottle and spill all its contents over the table if I do not close it immediately. But not because of any specific hygienic reason.
Pens... I clean them before switching inks, not every time I fill, with running tap water (good tap water here), and wipe them out if dirty before filling.
Nothing fancy. Normal inks have enough biocides to last a long (very long) time unless highly contaminated. At leas that has been my experience so far.
Occasionally, for very old bottles, I put some silicon grease on the lid or bottle mouth rubbing with a finger because dried ink can be a problem to open them in the future and I prefer to close them tight to avoid evaporation (ten years ago I had a bottle of black Cross ink -from the '80s- go so thick that it had trouble to run on some pens). So, I do not worry that much as you can see. Yet, my somewhat old Sheaffer, Parker Quink and Penman Ruby, MB Burgundy and some forgotten UK brand bottles first opened 30 years ago are still in working condition.
YMMV though: if there is high humidity, light and warmth, there could be a high level of fungi, bacteria and spores floating in the air and contamination might be easier. One should notice that though: stained walls, respiratory diseases, foul smells, etc... tend to (not necessarily) go hand in hand with high spore content, independently of ink.