(I am in mathematics, but similar language is used roughly similarly, I believe.) As a place-holder answer: a "principled" approach in science is at least opposite to a quick-and-dirty, or ad _hoc_, or "kludge-y" approach, the latter three synonymous expressions meaning that the priority is getting some result out, perhaps even finding some rationalization for the conclusion one wants. Obviously a non-principled approach more lends itself to corrupted (but also quick, desired, easy) results.
The "principled" approach "takes the high road", does not bias conclusions, does not rationalize-away weaknesses or flaws in methodology or information.
That is, one could hope that a "principled" approach involves no conflict of interest for the parties involved, and could be trusted. At its worst, "unprincipled" approaches (which no one would ever admit to, except perhaps as a mildly perverse claim to fresh unorthodoxy) produce completely untrustworthy outcomes, because those outcomes are chosen in advance, and whatever results are obtained are "interpreted" to support the original premise.
A hilarious example I witnessed was a computer science M.S. (details elided to protect privacy), on which I was an "outside examiner", in which "the goal" was to prove that two bunches of events were correlated, thus proving that the people who were promoting the one as "cause" of the other were right, and people should invest in their product. (Nevermind that correlation is not causality.) The guy failed to find any correlation in any of the first twenty or so statistical tests he applied... but he kept at it, until he found a statistical test that did seem to assert a slight correlation.
Of course, what he had really proven was that there was apparently no correlation... but, taking an "unprincipled" approach, claimed the opposite of what his own evidence showed, etc.