The question is not precise enough to get a definite answer, but not for the reason most people say in commentaries. The problem does not lie in the ambiguous meaning of "consistent" (which just means "free of contradictions", which was as clear then as now), but in the meaning of "way of arguing". What we do have is a corpus of results from the founders of calculus (say Pascal, Descartrs, Fermat, Newton, Leibniz), and a corpus of arguments they used to justify them. The corpus of results is certainly a corpus of true results, so is consistent, and was certainly recognized as such even by Berkeley (to my knowledge, the first serious contradiction involving results of calculus came 150 years after the founding period with Cauchy's theorem that a limit of continuous functions is continuous, combined with counter-examples from Fourier's theory, so is completely out of our scope).

Now is the corpus of arguments used by our fathers "consistent"? This question does not make real sense, because "arguments" are not results, and are not "true or false", either individually or in groups. They are, then and now, incomplete developments aimed at convincing one's that some results are true. The thing one can say is that, however shaky the arguments seem to us, they were used by these founders to prove only true results. In this very weak sense, their result was consistent.

Now, is the "way of arguing" of our fathers consistent  ? This again does not make sense, because there is no unique way to deduce from a finite set of examples was what the "way or arguing" of our founding fathers. What is sure is that a naive reader of their arguments, trying to guess, "by induction" in the sense of natural sciences, what was the way of arguing of these people, and trying to apply this way of arguing to get new results, would easily come across contradictions (even not so naive readers, such as Cauchy, eventually did so). Actually, it took almost 200 years for mathematicians to find a "consistent way of arguing" in which the arguments of the founder can be reformulated without too much distorsion: it is the $\epsilon,\delta$ approach of Weierstrass and others. It took almost one more century to construct a second consistent approach, which perhaps has the slight advantage on the classical one to reformulate with even less distorsion the arguing of our founders of calculus. Yet priority has a great weight in science, and this is the most obvious reason for which the non-standard analysis has not supplanted the traditional one.

I want to finish by a side, wittgensteinian, remark: we are not in a qualitatively different situation than our founding fathers webre : there is no way to be sure that our current "way of arguing" is consistent, because there is no way to be sure what our current "way of arguing" exactly is. By this I am not thinking at all at the problem that since Gödel we doubt that ZF or any other system is consistent, but to the much most basic problem that even with a "certainly consistent set of axioms" (say the axioms of the theory of groups, to fix ideas), we are not really sure what our way of arguing (that is the logico-formal rules which allow us transform statements into other statements, from axioms to theorems) is. To be sure, we mathematicians now take great care to begin a treatise by explaining carefully those "formal rules" or reasoning. Yet this formal rules use notions that are not completely clear (such as the notions of "intuitive integer") and skill that we can not be sure to posses (for example the capacity to recognize, in a finite expression, all occurrence of a given free variable). What we do is we see other people working using those rules, we try to do the same by imitation, getting punished if do it wrong, and after some times do not make mistake anymore -- so we deduce that we understand the rules as the others do. But there is no way to be really sure of that.

The question is not precise enough to get a definite answer, but not for the reason most people say in commentaries. The problem does not lie in the ambiguous meaning of "consistent" (which just means "free of contradictions", which was as clear then as now), but in the meaning of "way of arguing". What we do have is a corpus of results from the founders of calculus (say Pascal, Descartes, Fermat, Newton, Leibniz), and a corpus of arguments they used to justify them. The corpus of results is certainly a corpus of true results, so is consistent, and was certainly recognized as such even by Berkeley (to my knowledge, the first serious contradiction involving results of calculus came 150 years after the founding period with Cauchy's theorem that a limit of continuous functions is continuous, combined with counter-examples from Fourier's theory, so is completely out of our scope).

Now is the corpus of arguments used by our fathers "consistent"? This question does not make real sense, because "arguments" are not results, and are not "true or false", either individually or in groups. They are, then and now, incomplete developments aimed at convincing one's that some results are true. The thing one can say is that, however shaky the arguments seem to us, they were used by these founders to prove only true results. In this very weak sense, their arguments were consistent.

Now, is the "way of arguing" of our fathers consistent? Again, the meaning of this question is problematic, because there is no unique way to deduce from a finite set of examples was what the "way or arguing" of our founding fathers. What is sure is that a naive reader of their arguments, trying to guess, "by induction" in the sense of natural sciences, what was the way of arguing of these people, and trying to apply this way of arguing to get new results, would easily come across contradictions (even not so naive readers, such as Cauchy, eventually did so). Actually, it took almost 200 years for mathematicians to find a "consistent way of arguing" in which the arguments of the founder can be reformulated without too much distortion: it is the $\epsilon,\delta$ approach of Weierstrass and others. It took almost one more century to construct a second consistent approach, which perhaps has the slight advantage on the classical one to reformulate with even less distortion the arguing of the founders of calculus. Yet priority has a great weight in science, and this is the most obvious reason for which the non-standard analysis has not supplanted the traditional one.

I want to finish by a side, wittgensteinian, remark: we are not in a qualitatively different situation than our founding fathers were: there is no way to be sure that our current "way of arguing" is consistent, because there is no way to be sure what our current "way of arguing" exactly is. By this I am not thinking at all at the problem that since Gödel we doubt that ZF or any other system is consistent, but to the much most basic problem that even with a "certainly consistent set of axioms" (say the axioms of the theory of groups, to fix ideas), we are not really sure what our way of arguing (that is the logico-formal rules which allow us transform statements into other statements, from axioms to theorems) is. To be sure, we mathematicians now take great care to begin a treatise by explaining carefully those "formal rules" or reasoning. Yet this formal rules use notions that are not completely clear (such as the notions of "intuitive integer") and skill that we can not be sure to posses (for example the capacity to recognize, in a finite expression, all occurrence of a given free variable). What we do is we see other people working using those rules, we try to do the same by imitation, getting punished if we do it wrong, and after some times do not make mistake anymore -- so we deduce that we understand the rules as the others do. But there is no way to be really sure of that.