Shift Register Output tidied up by capacitor over a length of Cat5. What is happening?
It's kind of hard to guess from here, but your longer cabling probably was just picking up RF energy, and thus, high-frequency noise. The rather large capacitors are effectively shorting that noise to ground. Maybe you should use smaller capacitors and also some termination resistors close to the "receiving" pins.

Typically, I'd try to get one rather nicely shielded data+clock line (maybe, even differential) from whatever feeds the shift register to the display – and place the shift register close to the display driver.

Personally, I'd consider just getting a small microcontroller to directly put next to the display, and make that MCU receive data to be displayed – e.g. via UART, SPI or I²C, all of which are commonly found in MCU hardware – and directly talk to the display driver. Sure, a 74xx595 is probably cheaper than a microcontroller, but in my experience, displays are often mounted close to further user interface elements (LEDs, display background light, buttons and knobs), and it might be well worth spending 40ct – 1.50€ on a MCU that handles all that in one place, leaving only the business-logic relevant communication to be done over the long link. (cabling isn't free, either)