I don't quite understand you description, but I am afraid, you got it completely wrong.
First of all, common cathode displays are the less suitable for interfacing to a microcontroller, especially if they require higher voltage (have 3 and more LEDs in series per segment) which I assume is your case.
Second, if you have six 74LS373, it is probably not a multiplex. Also note, that 74LSxx parts are not rated for the high voltage required for the displays.
Third, you need something to limit the current to the segments (usually resistors in series with segments). You can use the internal resistors of the74LSxx outputs' "push-up" part for limiting up to 1-2mA, but that's a tricky and dangerous practice.
Fourth, the 74LS373 is not rated to drive the usual 10-20mA per segment - not per output, nor as a sum of outputs.
Also, I don't understand the role of the "Port" in your description.

The usual way how to use bigger LED displays in non-multiplexed configuration is the following:
- use common anode displays
- use a 5V power source for the logic and a V_SEG+V_{drop on series R} power source (often 12V for 3-4 LED segments) for the displays; both with common GND
- connect the + of the "high voltage" to the (common) anodes of displays
- use NPN transistors rated for the "high voltage" and the required segment current to drive the cathodes (one transistor per cathode=per segment). Often, transistor arrays are used (such as the ULN2803 and similar).
- connect current limiting resistors between the collector of transistors and the cathodes of displays (emitters to GND of course). When calculating them, observe the saturation voltages of transistors (e.g. typ. 0.8V-1V for ULNxxx, as they are Darlingtons), account for the V_SEG difference between LEDs of different color (if you have the) and if you use the decimal point/comma or colon, note that these usually have less LEDs in series.
- drive the base of transistors appropriately from the microcontroller - e.g. directly, or using parallel latches (e.g. 74HC573-4), using serial-in-parallel-out shift registers, I2C "expanders" etc.

You may want to design the drive circuit so that the LEDs are not ON during reset/startup as that might cause problems for not-so-well-designed power sources (and/or power supply paths).

The multiplexed setup is similar, except that the anodes are not fixed to the "high voltage" + terminal; rather via PNP transistors, driven again by a suitable means from the microcontroller, being ON one in a time. Current rating for both anode and cathode drivers has to be (re)considered and current li,iting resistor recalculated (reduced roughly by the rate of multiplex). It is advised to design an auto-switch-off hardware for case the multiplexing stops (e.g. due to software error)), as permanent higher current can damage the segment .

With common cathode displays the setup is similar, but "bottom-up" - you connect the logic power source and display ("high voltage") source at the + terminal (this might be impossible if GND and protective earth is connected at both supplies), connect GND of "display source" to all cathodes, use PNP transistors or arrays (latter is rare and more expensive) as anode=segment drivers etc...

Jan Waclawek