I don't know how seriously people took the 'threat' of home-taping at the time, but I guess the answer is 'not very' and it was a short-term preemptive strike in defence of the copyright principle. It benefited - or at least, didn't harm - the industry because it required at least one of your close-ish friends to own a copy and for you to buy a cassette (decent quality: approx £1-2) go round to tape it (or give them the tape and trust them to deliver). All for something that would regularly wear out or snap after a few dozen listens.
Compare with now, where your friend is the whole world and his collection is everything ever made, delivered in decent quality in a few minutes. And if you can't see the difference, then Guy Hands has a company he'd like to sell you.
By the way, as I understand, it's not the 'music industry' that's appealing for this legislation, it's the record industry. One has a future; the other, as things stand, doesn't.