At the most basic option, replace the existing controller with a basic BOB, and wire into the existing stepper drivers. The ribbon cable from the controller to the stepper drivers is what carries the step/dir signals.

You'll need to work out how you're going to handle the control voltage, as by the looks of it that machine uses 24V for inputs/outputs (some machines used 12V), and most cheap BOBs can only handle 5V.
To help with this, I'd make up a list of the all the wire numbers connecting into the existing controller, and then trawl the wiring diagrams and establish what they all do, making a note of what voltage they need/produce.

Upgrading the stepper drivers would give you the biggest performance boost, but the existing ones would be good enough to get the machine running, and you could upgrade later if you felt the need.
Upgrading the steppers won't give you much gain, as they're modern square type anyway.

The biggest problem I can see, is if those are two pots (Spindle Speed/Feed Rate Override?), getting some form of BOB/controller to accept the analogue input reduces options and bumps the price up.
No basic parallel port BOB will handle an analogue input.
A CS-Labs CSMIO/IP-M can handle two analogue inputs, however you would need to make sure it has enough inputs/outputs, as it can't be expanded. The next option up would be the CSMIO/IP-S.
The CS-Labs are good controllers, and fully support 24V inputs/outputs.
PoKeys57CNC has 5 analogue inputs, however it's designed to run with 5V inputs/outputs, so you'd either need to change everything to use the lower voltage (I'd personally stick with 24V, as it's far more noise tolerant), or add in buffer boards to step between voltages.
CNCdrive's UC300 series of controllers also have analogue inputs, however I think they're designed around 5V as well (I can't be bothered downloading the manual to check).

I personally prefer Dynomotion KFlop, but they only have one option for multiple analogue inputs (Kanalog add-on board), and it would be very much overkill for this kind of machine.

Off course, the other option is to use a basic BOB for all the main digital in/out, then use something else to handle the analogue inputs. Things like a PoKeys57E or U would give you analogue inputs, along with lots of extra digital IO, but again I'm sure they're all low voltage. Or if you like to experiment, you could even use an arduino to pass the information in via Modbus, but that can be a challenge to get working.