Tighten old 1960s doorknob
This is easy see the threads on the shaft ? You need to twist the  “hole cover” maybe after releasing a set screw at about 2 o’clock in the photo. Turning that cover clockwise will tighten it , 
if their is a set acres in that 2 o’clock position turn it the set screw or grub screw ccw and tighten the cover ,when  it is tight turn that set screw CW  to hold it tight. To me this hardware looks older and of higher quality than a 60’s house unless early 60’s , but this should help.