In principle, magnets and floppy discs didn't mix, any more than magnets and the stripe on your credit card do. In practice, it takes more magnetic flux than most people would expect to alter either. Magnetic fields fall off drastically with distance (inverse cube, not the usual inverse square) so you pretty much have to deliberately try to damage any sort of magnetic media with a handheld magnet.

Magnets were definitely bad for CRTs, because most of them depended on a lightweight metal shadow mask to keep the electron beam from each gun from illuminating the wrong phosphors. If you bring a magnet too close to a shadow-mask CRT, you risk magnetizing the shadow mask (a temporary condition that will eventually be fixed by the normal power-up degaussing cycle), or even warping it, which obviously ruins the tube.

Modern hard disk drives are reasonably tolerant of external magnetic fields. In fact, they contain strong magnets themselves. Again, only a small amount of physical separation is needed to keep those magnets from affecting the media.

Magnets have no effect on normal electronic circuitry, LCDs, or SSDs, unless (again) you go way out of your way to move a very strong one rapidly back and forth at a very close distance. You might be able to induce enough voltage in an unshielded inductor to damage something, if you tried hard enough. I wouldn't be surprised if it's impossible outside of an MRI machine room. Certain inductor cores won't take kindly to being magnetized, so there's some potential for damage there, I guess. Never seen it happen.