The wider spread of pop media, and its availability, I've seen over decades, has simply filled up so much space it has more and more crowded out exposure to anything else, accounting for the ignorance and also the unwillingness to even consider any music which is 'other' than pop.
Through the first half of the 20th Century, through some government funding, many a person who didn't listen to Beethoven was still aware of and liked the music of Copland. Copland was nearly a household name! There were national broadcasts of concerts, classical, newer contemporary classical, and not just Copland, with major conductors: all were 'on board.'
Later, Leonard Bernstein championed classical music (and contemporary classical music) in programming with the New York Philharmonic in a televised lecture-demo series on classical music aimed specifically at young people. And... in prime time on a major national network.
Bernstein had an amazing gift to get people interested, and not once seem like he was 'talking down' to anyone. Many an adult also eagerly followed the series, and became forever after more involved with classical music.
Now it does seems classical is more isolated, and dissed before even one tryout. I think unless one lives in a large cosmopolitan area where EARLY exposure to it is possible, even if accidental, it will remain foreign, distant and even alien to someone.
Fill up the air waves with pop culture, make it the overwhelming consumed thing, and on-line, television and all other media follow suit.
There is almost no decent public programming, and I include in that NPR (other than a few produced 'slot' programs which survive from a decade or more earlier, usually on or about one specific composer or Jazz or Folk.) Educational TV has changed similarly, not just in the area of music. Both are now so far off the mark as of general interest to other than a small demographic of people who are already educated and need no introduction, they can not possibly reach the average intelligent citizen who could be reached. A pervasive PC mentality has led them to a particular mind set, and that has nearly abolished all discernment of what is worthwhile which does not fit, limiting what is presented.
One of the nations' best FM classical stations has caved to the more than dreadful-ghastly 'theme' programming, at least for the a.m. rush hour (all upbeat baroque - and single movements of it) and a p.m. 'post rush hour' which is similar, but with slow and soothing music (again often single movements) 'to smooth out the listeners' day.'
With several generations used to pop music no longer than three-minutes listening span, music which already repeats its material wholesale at least three times within that three minutes (you've really only 'followed' new material there for one minute,) getting anyone raised on that to sit still for the six, eight or more minutes a classical piece takes to introduce itself and warm up to the main business, can take some real effort on those so conditioned. it can initially fatigue their brain and then they will not give the classical any more time. Listening to that short-end material for a lifetime, and presenting it then to the next generation as 'classic' in no way prepares ones attention span for what is needed to follow most classical music.
People now seriously think the likes of John Williams, Danny Elfman, Hans Zimmer and Video Game sound track composers like Joe Hisaishi ARE classical composers and that IS classical music! To them, those are "the greatest classical composers of today."
Cost is not an argument: Pop CD's and concerts to live events cost the same at least - attending a live pop concert often more - than a very decent seat in a hall to hear a world-class orchestra.
Best regards.
ADDED: With the resounding slap of my esteemed colleague Nemesis truthful answer still stinging my pate, I confess - partially - to caving into the elitist doom and gloom atmosphere this question sets: I delineated many a contributing factor to the present "American Cultural Condition,' and much of it, imho, is true. Resistance from younger people to classical here has much to do with a real disconnect from the European past + a pervasive and strong anti-intellectual climate. Otherwise open-minded people here have greater negative association with Classical as being 'guys in powdered wigs and breaches' - remote and irrelevant to any person alive now. It takes more to break through. I have successfully introduced many young people to classical who were immediately engaged and who remained engaged. It took an absence of attitude and thoughtful selection of repertoire, that's all. It may take more care to introduce an inexperienced young American to classical music, but not much more than careful thought, a bit of awareness of the nature of their resistances, to put them at ease enough so they listen with open minds.