As phoog's answer notes, most (all?) tuning instructions you find in 17th and 18th century sources tend to be given in terms of actual pitch names, where specific intervals between specific pitches were tuned precisely. So, it's not so much, "Start on your tonic note and tune these intervals around it," and more like, "temper the fifth from E-B like this, and the one from A-E like that," etc.
However, the effective "key center" of most of these historical temperament instructions tends to be around keys with few sharps or flats. So, if the question of the "base note" here means something like "What key/keys were most in-tune for historical temperaments?" then keys like C major, G major, A minor, etc. were generally least likely to have intervals that sounded awful. The more black keys that appeared in a particular piece, generally the more likely you'll start to veer into intervals that sounded less in-tune. (This is a broad and very general statement, as temperament systems varied a lot in how they were designed.)
I suspect this question is generated by the feature that appears on many modern electronic keyboards where one can choose a "center note" for the particular tuning, like selecting an A♭ center pitch for a piece written in A♭ major. What this effectively does on most keyboards is to transpose the tuning system so that A♭ and neighboring keys tend to be most "in-tune," rather than C or whatever. This process is very ahistorical, though it can be fun to play around with on keyboards that have such a function.
I do seem to remember reading somewhere about composers or performers making adjustments to tuning to accommodate different keys. So, one might choose to temper a different set of fifths according to the instructions instead of the standard one. Or perhaps just use a different tuning system altogether to accommodate playing in more sharp keys or more flat keys or something. I can't recall where I read this off-hand, but it certainly wasn't commonly practiced in performance or anything. Maybe a performer who knew the pieces on a program had a wider selection of keys or something might choose a more "forgiving" temperament when tuning the harpsichord in preparation for a performance, but I don't know off-hand if we have any accounts of this.
The other thing to remember before going too far down the temperament obsession rabbit hole (something many people have done, myself included) is to recognize that these complicated sets of tuning instructions were generally only applicable to keyboard instruments like harpsichords and organs. When you start involving varieties of other instruments in ensembles and singers, suddenly there's often going to be a much greater variety of tuning issues. And you'll effectively have several different simultaneous "base notes" for tuning. The trumpet or horn (using natural harmonics) pitched in whatever key combined with a flute or oboe pitched in another key combined with the strings (with their Pythagorean tuning tendencies) will suddenly make a lot of those "a few cents here or there" tuning issues for the keyboardist less relevant. Particularly once you get to the 18th century and had circulating temperaments that sounded "passable" in all keys.
In general, I think what the question is getting at is whether and how performers adjusted instruments to make the keys of their programs sound "more in-tune" by changing the temperament. And the general answer is that once a large variety of standard keys became common (with several sharps and flats), the notion of the "characters of the keys" started to emerge, probably at least partly due to the differences in which keys sounded more or less in-tune with common temperament procedures. (And this could be exploited in composition: I remember encountering a passage in a Handel recitative that rapidly modulated through some unusual keys, stopping for a while in A♭ minor on the words "horrible, horrible." Yes, the accompanying harpsichord probably did sound a bit "horrible" during that passage, which likely was the point.) Reorienting your tuning system to accommodate music of different keys would likely undermine these tendencies.
