Does anyone believe the Nina Kuligina videos from the 70's are legitimate? If so why, if not why not?

Of course they are not legitimate, if anyone really had amazing telekinetic powers there would be better evidence than fuzzy old black and white videos. 

She is very wiggly for someone that is supposedly moving things using only her mind.

If the quality of the recording was better you could probably see the strings. 

Nina Kulagina was claimed to possess 'psychic' and 'Psychokinetic' powers. You should be immediately skeptical given it is widely known that science has never shown such powers to be real. You should be further skeptical because there is no scientific plausibility for such powers. But most of all, you should be skeptical because we all know there are many frauds out there.

If the question is are the videos themselves legitimate, yes they are. But they don't validate psychokinetic powers in any way.  Extraordinary claims require extraordinary evidence. 

Kulagina was suspected of utilizing hidden magnets and threads to perform her feats. She was caught cheating on more than one occasion (reference Joel Levy, Mike Dash and Martin Gardner)

I hope not. It would be very sad that were true.