Because the real numbers are uncountable and the real algebraic numbers are countable, there are uncountably many transcendental numbers. So there are far more transcendentals than rationals. With the language of set theory these facts are easy to see, but historically it was very difficult to construct transcendental numbers (or prove that certain numbers were transcendental). It is my impression that, historically, nowhere-differentiable functions were considered pathological and rare among the continuous functions. Was this similarly true for transcendental numbers among the rationals: were they considered pathological and rare?
As I understand, people did not think much about comparing the size of infinite sets before Cantor, so for most of them the question you are asking had no meaning, and they did not think about it. Perhaps few people felt intuitively that a countable set is "smaller" than a continuum, but Cantor was the first to state and prove this.
Edit. But there were actually exceptions to this. Nicola Oresme in 1360 (!) really argued that there are much more irrational numbers than rational ones, and even that a "random number", whatever it could mean, is irrational. For more details on this see the answers to this question https://mathoverflow.net/questions/269893 and references there. Of course his arguments could not be rigorous at that time. And he was discussing irrational numbers. That transcendental numbers exist at all was only proved in 19 century, by Cantor and Liouville, with different proofs.
is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x*x − 2 = 0.
Thus, with insight, now that we know Cantor's proof of the uncountability of the real numbers, we can easily conclude that we have a lot more transcendental numbers than non-transcendental ...
But this is the history made from the point of view of modern time; as explained by Alexandre in his answer, at the time of Liouville's proof of the existence of transcendental numbers (1844) there was no clear understanding of the possibility (if any) of comparing the "size" of infinite collections [you can see this post].