While I cannot talk about all of Europe, I can talk about Greece. I feel it is useful because it illustrates that the reception of the Balfour Declaration then, (and now but I digress), was not always as straightforward affair as one would expect. Others can offer examples of other countries but I feel the Greek paradigm has to offer an interesting case of how conflicting attitudes can co-exist.

In 1917 Greece is in the last phase of a semi-Civil war where the pro-Entente nationalist liberal PM Venizelos pitted himself against the pro-Central Powers King, in an effort to draw Greece into WW1. Venizelos in the ends wins, after having previously established a separate goverment at the freshly conquered Salonica, with the military help of the Entente. Salonica itself is the second most important city of the Balkans after Constantinople and it is only five years since 1912 when it passed into Greek hands after the Balkan wars which obligated the Ottoman Empire into relinquishing almost all of its European possessions to Bulgaria, Greece and Serbia. Salonica coveted by Bulgarians, Greeks and AustroHungarians is home to a millenia long Jewish community unlike the ones found in Eastern Europe. The Jews are indigenous to the city and constitute the majority of the population creating a fully fledged Jewish society with links to Europe. It is confident enough to demand a say in its future and even dreams of self rule, perhaps in the form of a Free Port. The community politically is divided in three different factions although in 1917 we find them in their earlier proto-form: the assimilationists, the zionists which are a vocal but small minority and the Communists.

Greek rule over the city is not a given. Venizelos' "Goverment of Salonica" in the beginning and national goverment afterwards was made possible only through military intervention by the Entente. As we said Bulgarians primarily and AustroHungarians secondarily have plans about the city. And while the first do not enjoy any local support, they are militarily a major threat. Greece knows that everything is still up for the grabs and it while defending its gains, it still has eyes for more territory where huge Greek populations reside. That's why Venizelos is geared to enter WW1, since he believes in an Entente final victory.

Greece bases its claims of geographical enlargement on historical rights, being a responsible conqueror which merits being an ally of modern Western powers and of course the right to self-determination of the Greek populations of the Near East. All these reasons combined with the politically advanced Jewish population and the spotlight under which Salonica is by virtue of being the center of one of WW1 fronts, make Greece extremely eager to appease its Jewish population.

Even better Zionism helps not only appease the Jews but hurts the Ottoman Empire plus it physically removes Jews from the newly conquered territories removing social and financial competitors.

Greece, or at least the Goverment of Salonica, recognizes Zionism even before the Balfour Declaration in July 1917 with PM Politis stating so in a local salonican newspaper. As far as I know this is the first official state endorsement of Zionism in the world. Later having established the general control of Greece, the now unitarian Greek central goverment acknowledges and adopts the Balfour Declaration.

This acknowledgement is not just empty words. The first Zionist Congress of Greece is warmly received by the government which even instructs local prefectures to give in every Greek city with Jews a street named "Palestine Street" in order to expressly show solidarity with the Zionist cause. Zionist flags are proudly displayed in the streets of Salonica, alongside Greeks one and with noone noticing any incongruence.

So we see that Greece is an enthusiastic supporter of the Balfour Declarations since it aligns with every single goal of Greece: weakening of the Ottomans, self-determination of the people, homogenization of Greece by removal of non-Christians and appeasement of a powerful Jewish population in the most important of its newly conquered territories.

All this contrasts with stark religious antisemitism in which Greece in still imbued. The 1891 pogrom of Corfu is not exactly ancient history and neither the false accusations of treachery of Jewish soldiers during the previous war of 1897 which the Ottomans wond handily. Each Easter Jews/Judas mockups are being burned in Greece and the blood libel reigns supreme. Attitudes which will surface again after 1923 when Greece has no longer need of appeasing the Jews and lets prejudice prevail.

tl;dr So Greece is a good example of problematics which arise when responding to the question. Greece is undoubtedly antisemitic and we can see this before the Balfour Declaration and after the Balfour Declaration. But for a brief window between 1913 and 1923 Greece is arguably one of the most pro-Zionist and pro-Balfour countries in the world since its political aims are marvelously aligned with the zionist goals. So in this case antisemitism not only did it not hinder the adoption of the Balfour Declaration which was for national/politic expediency but also found itself an ardent supporter since it lead to a Jew-less Greece.