Yes and no.
Yes, in that the basic notion of cardinality is fairly important, in particular whether a set is countable or uncountable. If it's countable, you can parameterize all the elements as arbitrary-length finite strings on some finite alphabet. If it isn't, then you can't, and you need to either look at infinite strings, successively better finitistic approximations, or truncated infinite strings ending in "dot dot dot." This basic question arises commonly enough in most fields of both pure and applied mathematics that it's something you ought to know.
No, in that beyond the above fairly basic question of countability, transfinite quantities don't seem to get involved all that often. Almost all of applied math can be formalized using only three infinite cardinals - that of the natural numbers, that of the reals, and every so often, that of the power set of the reals. More to the point, most of the most powerful applied math techniques we have involve finitistic approximations of problems, such as in numeric computing, so that we are only computing things to arbitrary accuracy anyway and infinity doesn't really get involved in any meaningful sense.
The essence of the problem, unfortunately, lies in that the theory of transfinite cardinals and ordinals hasn't quite "developed" in a concrete way, similar to the rest of math, as Cantor hoped it would when he first created it. We cannot prove even basic things like if the reals are the next-largest cardinal after the naturals (the continuum hypothesis), which we now know, thanks to the results from Cohen's celebrated landmark paper on "forcing," is independent of ZFC set theory. In general, the structure of the cardinals is heavily dependent on the set theory, so to some extent you can "choose your own adventure" - yet there is no consensus on how to choose or even if there is a "right" way to choose.
As a result, it's difficult to use transfinite numbers as a core technique in applied math when much of the very basic questions are still open and buried in murky set-theoretic foundational mess.
The theory of ordinal numbers, on the other hand, does not seem to run into quite as much set-theoretic trouble right out the gate. I haven't seen them used that much in applied math (although certainly on occasion), probably because of the emphasis always being on numeric computation, which typically involves finitistic approximations of everything anyway.