Again, looking at the results from BCCWJ
方がいい is "correct", at least descriptively speaking, although the results also suggest that if you're going to write 方 in 漢字, you might as well write 良い with 漢字 as well.
As for the difference between 漢字 with the use of nouns and verbs, we can check for nouns
and for negated verbs (and negated i-adjectives)
This looks like 方 is slightly more often written in 漢字 when accompanying a noun (この方 and その方 account for 51 results), and sightly more often written in かな, when accompanying a verb.
(And lastly, そのほう has 53, その方 45 results.)