Simply said, we have reached a moment in Western history when, despite all appearances, no meaningful public debate over belief and unbelief is possible. Not only do convinced secularists no longer understand what the issue is; they are incapable of even suspecting that they do not understand, or of caring whether they do. The logical and imaginative grammars of belief, which still informed the thinking of earlier generations of atheists and skeptics, are no longer there. In their place, there is now—where questions of the divine, the supernatural, or the religious are concerned—only a kind of habitual intellectual listlessness.
True enough. And Christians likewise have reached similar impasses in the past. You could tell those impasses had been reached because the dialog between the Church and the culture reached the point where the culture was crucifying, burning, shooting, and hanging Christians. The solution of the Church to such impasses (offered by Jesus) was "Do not give dogs what is holy; and do not throw your pearls before swine, lest they trample them under foot and turn to attack you. (Matthew 7:6) Jesus' counsel to the Church is not to beat your head against a wall, but "wherever they do not receive you, when you leave that town shake off the dust from your feet as a testimony against them.” (Lk 9:5–6)
So if our elite are so sodden with intellectual ennui that they won't listen, the obvious thing to do, I think, is do what Paul did in similar circumstances:
And Paul and Barnabas spoke out boldly, saying, “It was necessary that the word of God should be spoken first to you. Since you thrust it from you, and judge yourselves unworthy of eternal life, behold, we turn to the Gentiles. For so the Lord has commanded us, saying,
‘I have set you to be a light for the Gentiles,
that you may bring salvation to the uttermost parts of the earth.’”
And when the Gentiles heard this, they were glad and glorified the word of God; and as many as were ordained to eternal life believed. And the word of the Lord spread throughout all the region. (Ac 13:46–49)
"But our intellectual elites have a lot of influence!" Yep. Which means we don't stop talking to them and about their ideas (some of which are good and some terrible), but we also don't make conversion of their cold, dull, and uninterested hearts the criterion by which our evangelistic efforts stand or fall. Instead, we recognize that they offer a counter-gospel--and a dull and boring one it is: a gospel so boring even they aren't interested in it. Then we present the joy and glory that Christ offers to the people in our culture who will listen. There are still lots of them, as the thousands of people who just entered the Church at Easter attest.
As Fr. Robert Barron points out, this will require a new mind and spirit from us Catholics, but it can be done, as he himself is tirelessly demonstrating: