I am more versed in music, so if we take Mahler as an example - were there any specific, tangible, criteria by which his music was deemed to be Degenerate?
The Entartete Musik exhibitions did have particular criteria, but you're quite right in that simply being composed by a Jew would be more than enough to meet the criteria.
The Reichsmusikammer or Reich Music Chamber compiled specific lists of composers that were banned, as well as entire groups of composers and compositions, which included all music composed by anyone Jewish, black, Roma (Gypsy), Gay. As well as all Jazz and Swing, which was labelled as Negermusik (not translating that one - I assume you can figure it out) and so anything Jazz was banned, even if the composer was white and German. Serialist/12 tone music was also banned, as well as music by Americans, French (except Bizet), Russians, Poles (except Chopin) and most foreigners. This includes not only composers, but also performers. If you were Jewish, you would be blacklisted, even if you were performing German works by Wagner.