There are no strict criteria but we have some guidelines. [adj-no] is inherently problematic, as Sombrero1 showed in his excellent comment. Please see the most recent comment on 南極地方 where I discuss the tag in some detail.
Sombrero1 is correct. This is not [adj-no] as we currently use the tag.

Then what criteria should determine whether a sense gets an adj-no tag? Is it based on corpus frequency, lexicalization, idiomatic English equivalents, or some other principle?

> There are clearly uses of 家庭の〜, so it works as a の-adj in function.
This would imply that every N + の + N construction is adjectival in nature, which isn't the case.

の first and foremost indicates genitive case, making it quite a bit more difficult to draw a line in between "[adj-no]" and "not [adj-no]"
From what I've seen so far the [adj-no] tag is heavily dependent on the perceived degree of adjectivality, which is then often mirrored by an adjectival translation into English (take 常磐 as an example). 

While I could see "家庭の" translating as "domestic" in some cases, that seems to be about it.
In the ngrams roughly 10% of hits are 家庭の～, which isn't overwhelming.
Most collocations in the top100 ngrams don't really seem that adjectival to me. Some of the most common are:
家庭の医学 (also name of a book), a fairly standard genitive N + の + N construction
家庭の事情 (apparently also name of a film), same thing as above
家庭の教育
etc.

https://www.edrdg.org/~jwb/cgi-bin/ngramlookup?sent=家庭の&topjuku=on&top100=on


Furthermore, NINJAL corpora don't parse 家庭 as a 名詞-普通名詞-形状詞可能, which they do in cases like 空っぽ (coinciding with our tagging).
There 家庭 is simply a 名詞-普通名詞-一般 in all instances.
IIRC NINJAL uses Unidic as well


There is certainly no objective ruling to be made here, at least not on my part, but to me this does not seem like an [adj-no] either.
And if there are cases where 家庭の is indeed of adjectival nature (which I don't doubt), it doesn't seem substantial enough to warrant the tag.

家庭のテレビ	12783
Home television (family tv at home, not a genre of tv)

There are clearly uses of 家庭の〜, so it works as a の-adj in function.
All の-adjs are simply nouns that describe another noun using の.

I disagree with the idea that 家庭の is an adjective. "home" in "home cooking" is not an adjective either.

@@ -18 +17,0 @@
-<pos>&adj-no;</pos>
@@ -23,4 +21,0 @@
-</sense>
-<sense>
-<pos>&adj-no;</pos>
-<gloss>homemade</gloss>