As pointed out in the comments to this question, there can't be a definitive answer because genres are a fluid thing. I wasn't aware that some commentators regard the idea of a link between the real world and the magical one as archetypal of high fantasy, and I'm not sure why,as for me at least, the distinction between high and low is about the dichotomy between a focus on heroism or realism, either of which can be present regardless of a real world link.
I would say that the difference between high and low fantasy is tonal. High Fantasy takes its inspirations from the historical epics, by way of Tolkien, which of course drew on classical literature heavily for many of its tropes. Low fantasy is more gritty, and in some sense a response to the limitations of the kind of stories that can be told in the high fantasy mould. It involves elements of the lives of ordinary people, moral compromise/shades of grey, and a low key and/or low magical setting. The two genres are not entirely distinct and some works may have elements of both.
Lewis was writing at the time of Tolkien, and was in fact friends with him, so it would not be surprising if there were some stylistic similarities. I would argue that Narnia is in fact high fantasy because
- the narrative focuses on mighty world changing deeds in the epic style and the central characters are frequently heroic or villainous in an uncomplicated way
- Magic is commonplace - White Witch, Aslan, talking animals
- The key characters frequently are or become rulers
There are, however, some factors that might support the idea of low fantasy
- some parts of e.g. "The Lion, the Witch and the Wardrobe" focus on the plight of the common people under a tyrannical witch vastly superior in power to them
- arguably there aren't that many actual magic users in Narnia - although the talking animals are in some sense magically imbued by Aslan, there is a sense in which that's more a part of the background to the world than active magic within the context of the story.
- Much of "The Magician's Nephew" in particular focuses on characters without magic or power caught up in a struggle with a force beyond their comprehension (although this does lead into the epic events of the creation of Narnia).
On balance, I'd say that the epic/mythical nature of the Narnia stories places them more naturally within the realm of High Fantasy, and perhaps this in unsurprising given the historical context in which it was written.
