I'm going to assume that by non-synthetic, you mean natural "cork" corks. But, even if it's natural cork, it's usually not a big concern, especially with the length of time beer is stored for.
There's a lot of humidity in the bottle already which should keep the cork from drying out, although on a related note, you want to make sure the air isn't too consistently dry OUTSIDE the bottle. 20% and up most of the time should be sufficient. A fridge will dry it out though. If need be, you can always get a cheap humidifier for your cellar or wherever else you store your beer.
Natural corks can also introduce musty flavor characteristics to the beer and cork taint when touching the beer, which might be okay for some farmhouse ales, but is generally a bad thing.
Also... when you lie the beer flat, you are making it shallower but wider. The "wider" part is the problem, since it means much more of the beer is directly in contact with the air above it, leading to much quicker oxidation.
Basically, unless you're planning on storing that barleywine for well over a dozen years, there's pretty much nothing to worry about. Even then, I wouldn't be concerned. The environment you store it in is the important part - a cool temperature, decent humidity, and limiting UV exposure (keeping them covered or in a box, avoiding fluorescent lights in that room and *ideally* using LED bulbs) are the things to really focus on.
As for the natural corks... I try to avoid those when I can, mainly using them for Bière de Garde, where I *do* store them sideways for at least a month or so.