I can only address Catholicism, as I have not sufficiently studied how Protestants in Germany responded to the challenges of Nazi rule to adequately describe them.

In Catholicism there is indeed a tradition of anti-Semitism. In the 19th and 20th centuries there was an increase in official support for a distinction between anti-Judaism (at its very basic an assertion that Judaism is the wrong religion) and anti-Semitism (the racial component), and that anti-Semitism could not be supported by Catholics. For instance, in Mit Brennender Sorge (March, 1937) section 8:

Whoever exalts race, or the people, or the State, or a particular form of State, or the depositories of power, or any other fundamental value of the human community - however necessary and honorable be their function in worldly things - whoever raises these notions above their standard value and divinizes them to an idolatrous level, distorts and perverts an order of the world planned and created by God; he is far from the true faith in God and from the concept of life which that faith upholds.

sections 15 and 16 praise the Old Testament, an oblique statement of support for German jews. This becomes more explicit in section 18:

The Church founded by the Redeemer is one, the same for all races and all nations. Beneath her dome, as beneath the vault of heaven, there is but one country for all nations and tongues; there is room for the development of every quality, advantage, task and vocation which God the Creator and Savior has allotted to individuals as well as to ethnical communities

Another explicit refutation of the ideology of race is present in section 11:

None but superficial minds could stumble into concepts of a national God, of a national religion; or attempt to lock within the frontiers of a single people, within the narrow limits of a single race, God, the Creator of the universe, King and Legislator of all nations before whose immensity they are "as a drop of a bucket" (Isaiah xI, 15).

The call to action by individual Catholics is in section 19, "at no moment of history, no individual, in no organization can dispense himself from the duty of loyally examining his conscience, of mercilessly purifying himself, and energetically renewing himself in spirit and in action."

In October of 1939 Pius XII published Summi Pontificatus. WWII had just begun in Europe, but the papal message was consistent with Pius XI's Mit Brennender Sorge. There are six mentions of the "human race"--an appeal to unity in a divided Europe--along with other appeals to racial unity in sections 45, 52 and 72. Section 48 points out the mixed racial origins of Christ's apostles as well. Finally in section 106 there is a denouncement of attacks on noncombatants, particularly referring to Poland. This theme was continued in the pontiff's 1942 Christmas address.

Mankind owes that vow to the hundreds of thousands of persons who, without any fault on their part, sometimes only because of their nationality or race, have been consigned to death or to a slow decline. Mankind owes that vow to the many thousands of non-combatants, women, children, sick and aged, from whom aerial warfare—whose horrors we have from the beginning frequently denounced—has without discrimination or through inadequate precautions, taken life, goods, health, home, charitable refuge, or house of prayer...

It is clear, sadly, that many Catholics had divided loyalty and did not heed papal statements regarding anti-Semitism being unacceptable for Catholics (among other items that were unheeded by Catholics). Far too many chose country over Church in this instance, and many Catholics participated in the atrocities of WWII. The Catholic Church's tradition of anti-Semitism made it more difficult for Catholics to react to papal calls for a change in policy, and made it easier for Catholics to be anti-Semites. This is a tragedy in my estimation.

You can find my longer statements on Pius XII here. As always, followup questions from OP and others are highly encouraged.