In what language was the first Zionist congress in Basel held?
Was it Yiddish, Hebrew, German, English? Where there translators?
This is an excellent question and this answer is only the "easy" answer based on easily available sources, and should be used primarily as a jumping off point for more research on what is in fact more likely a more complicated reality.
The full PDFs of the stenographic protocols of the Zionist congresses from 1897-1935 are available here:
All of these transcriptions of the speeches at the congresses are in German, but the 1897 congress, alone among all them, contains the protocol in both Hebrew and German:
I think it is safe to say, however, that the main language of this congress too, was German with a Hebrew translation of the protocols added. Skimming through the protocol, the majority of the speakers in the congress are marked as coming from Zürich, Köln, Berlin, Bingen, Wien, Frankfurt, Prague etc. where German would be the primary language. Most of those who were not from a German speaking area, very likely knew German:
A few others among the participants you might want to check on: Adam Rosenberg (New York), Shepsel Schaffer (Baltimore), Jacob Berstein-Kohan (studied medicine in St. Petersberg, perhaps his letters to Weissmann will give a clue).
Also, the invitation card, and the programm for the conference were in German:
Also, the most famous two addresses, by Theodor Herzl and Max Nordau are usually translated from the German, which would be unusual for such important documents, if they were originally delivered in Hebrew or Yiddish.
Probably More To This
I think that even if the main language or official language was German, when you bring together something like 200 delegates from nearly two dozen countries, the actual experience was likely to be much more complex. Through a process of purification through editing, the language of the protocol very likely hid serious code-switching, the insertion of Yiddish or Hebrew phrases, and other linguistic mixing that is common in these kinds of settings.
Marcus Ehrenpreis gave a talk on the Hebrew language. He grew up writing Yiddish, and it wouldn't be surprising if Yiddish made its way into his speech. Jacob Berstein-Kohan may have used French while studying at St. Petersberg and he could probably assume, if a German word didn't come to mind, that dropping in a bit of French now and then would be fine. Of course, this doesn't come through in the record, but may come through in diaries or memoirs if you continue research.
One place to start would be the University of Basel, where there was a 1997 exhibition on the congress:
Der Erste Zionistenkongress von 1897: Ursachen, Bedeutung, Aktualität: "... in Basel habe ich den Judenstaat gegründet. " Hg. von Heiko Haumann u.a. Basel 1997 [Begleitpublikation zur Ausstellung].
The First Zionist Congress's official language, both spoken and written, was German, but many delegates also spoke Yiddish (Hebrew-German vernacular), the language of Ashkenazic Judaism, and a Yiddish-like German known as Kongressdeutch.