(Migrated from comments at request of OP.) The question is a bit difficult to answer without knowing how the topics (e.g., infinitude, irrational number) have been defined. Nevertheless, since the question does not broach countability and is strictly about explaining the infinitude of irrational numbers, one might proceed as follows:
Show (or recall) that the sum and difference of any two rationals is rational.
Observe that the sum of any irrational and any rational is irrational. To prove this, suppose $a$ is irrational and $b$ is rational. Now: If $a+b$ were rational, then we could write $a = (a+b) - b$, i.e., we could write $a$ as the difference of rational numbers, hence $a$ would need to be rational; a contradiction. Thus, it must be the case that $a+b$ is irrational.
Show (or recall) that $\sqrt{2}$ is irrational.
Using #2, observe that $\sqrt{2} + q$ is irrational for all $q \in \mathbb{Q}$. This gives "infinitely many irrational numbers" as desired. If preferable, you could even modify and use $\sqrt{2} + n$ for all $n \in \mathbb{N}$.
But since this is MESE (not MSE): Ask students what they have already tried. Have they seen the proof that $\sqrt{2}$ is irrational? If so, has anyone suggested $\sqrt{p}$ is irrational for all primes $p$? Maybe someone has even gone and proved such a thing. Has someone suggested $\sqrt{n}$ is irrational for all natural numbers $n$? If so, where does such an argument break down? (And why can it be rescued by looking at, e.g., the square roots of primes?) If someone has tried the trick suggested above: Can they prove the different building blocks? Can they state anything meaningful about the sum of two irrationals? Can they solve related problems (e.g., given an irrational number and a rational number, prove there is always an irrational number strictly between them)?
Of course, if the irrationals have been defined in some other way, then your explanation may differ. For example, if one defines them as the non-repeating decimals, then can they prove $0.1234567891011\ldots$ is irrational? If they can prove this does not repeat, then can the proof be adapted to show the same about $1.1234567891011\ldots$? What about $2.1234567891011\ldots$? And so forth.
You write that your job is to help the students in this Calculus course to solve the problems, and that this particular one showed up. I am not convinced that showing them "an easy way" to answer this problem is the best approach with regard to helping them further down the road. Let them struggle with the problem; let them spend some time with the problem; demand to see some evidence that they have given the problem thought before you start to scaffold and assist them in figuring out a solution!
