In the consumer realm, you went to the store to buy your TV or computer and brought it home in your station wagon, possibly bribing your neighbor with beer if you needed help with the big heavy boxes or just wanted some company for the ordeal.
In the business realm, companies like IBM and Dell had (and still do) their own carriers that they contract with because it gives them more control over every part of the delivery process.
That said, CRTs are not particularly fragile, they are just heavy and bulky. With enough Styrofoam, they ship just fine. I did it multiple times. It costs a lot more than you'd like, of course, but generally cheaper than a plane ticket.
I was actually surprised when I learned for the first time that Amazon sold big LCD televisions. They are MUCH more fragile than a CRT on account of being a big pane of thin glass that's not allowed to bend or twist at all. I have to imagine the shipping loss rate of these is somewhere in the single digits at least.