Edit. I looked briefly at Cohen's original PNAS 1963 article, and for much of itin that article, he does not use the forcing-downwards notation at all. Rather, he uses the containment symbol $\supset$ explicitly. Thus, the assumption in the question that Cohen did indeed use the downward-oriented relation may be unwarranted. Probably (Perhaps this view is a little softened by the downward version observation that he consistently uses $\supset$ rather than $\subset$.)
Here is my theory. In logic and set theory there has been a long-standing tradition of consistently using the relation ≤ in preference to ≥, presumably to avoid the problems associated with mixing up the greater-than less-than order. Perhaps this goes back to Cantor? Now, in the case of forcingcame along , it is usually the case that you have a condition P already, and you want to ask whether there is Q stronger than P with a certain property (one rarely asks for weaker conditions this way). Thus, if you have the downward-oriented relation, you can economically say "there is Q ≤ P such that..." This is just how Cohen's text reads, since he says "there is Q \supset P such that ...". Generalizing Cohen's containment order to an arbitrary partial order, one then wants to interpret containment as ≤. And then the further support for this convention arrives with the fact that it agrees with the Boolean algebra approach to forcingorder a few years later, which arrived shortly after Cohenso it became standard (except for the Shelah school and a few others).