Two things usually drive the decision to include/exclude a phrase or sentence from translations of the NT:
1) An underlying analysis of the 'family tree' of the hundreds of thousands of fragments & manuscripts of the NT text, which is neatly summarised in a short table on Wikipedia's Textual Criticism page. In brief, all the translations in your list are 16th–19th century European, and based largely on the "Byzantine" text-type which was available in Europe at the time.
Modern translations—and the dividing line is the publication of Westcott & Hort (1881)—have preferred the Alexandrian text-type which is (if I remember correctly) about 10% shorter. One reason for this switch is that during the 1800s western scholars got access to older, 2nd–4th century, evidence for the Alexandrian type, compared to 5th century onwards for the Byzantine. (How these dates are arrived at is another subject). Earlier European translators only had access to the Byzantine text types.
2) A comparison of the various manuscripts in any given case. For this, without travelling round the world's universities, musea & libraries, we usually turn to "the" critical edition of the Greek New Testament, the successor to Westcott & Hort, which is the Nestlé-Aland and/or the UBS Greek NT (they are nearly identical) which omit the verse. (Sadly, unlike the print edition, the website does not list which specific manuscripts include or exclude the verse).
The UBS print edition gives this same list—@the-nontheologian's answer gives most of the list—and then adds a rating from A (which they call "certain") down to D ("which occurs only rarely, indicates that the Committee had great difficulty in arriving at a decision") at places where there is manuscript variation. For Acts 15:34 they give an A-rating to omitting it. (For short, often one-sentence, comments on why the UBS committee rated a passage as such, Bruce Metzger "Textual Commentary on the Greek New Testament" is the source).
The reasoning used when some manuscripts include and some exclude a passage is sometimes one of:
There is a repeated word or phrase in the longer version of the passage, and the shorter versions seems to skip over the words in-between. We then guess the shorter one is a copyist's slip-of-the-eye, and the longer version is original.
The longer version seems to be an expansion or explanation of the shorter version, and the shorter version seems quite 'rough' or hard to understand. We then guess this is a copyist's deliberate addition, trying to explain a difficult text, and that the shorter version is original.
And this brings us to the second reason that modern translations (where 'modern' means, after Westcott & Hort) prefer the Alexandrian text-type. It's because many of the differences between Byzantine & Alexandrian—including exactly your example of Acts 15:34—are more easily explained by option 2: that the longer text is a copyist's explanatory expansion of a shorter original text. See e.g. Wikipedia page for Alexandrian_text-type.
In the case of this verse then. We (1) look at the manuscripts which include/omit the verse (2) notice that the usual two arguments given above in favour of Alexandrian text-types both seem applicable here; and (3) come down on the side of the Alexandrian text-type and say that v34 is an explanatory gloss, not the original text.
And that's why modern translations all omit it.
