I don't think this is a Safari issue (as long as you have cookies enabled). I'm a longtime Citibank online customer, and the last time Citibank revamped their website, I began to have problems with using Safari to access it. They appear to have optimized their citi.com and citicards.com sites for Chrome, and then added support for other browsers on top of that... they don't seem to be investing much effort in having it work smoothly for Safari.
I just use citi's preferred browser and it works fine. Slow and clunky, but fine.
(I wish they would optimize it for Safari!)