Some Thoughts ...
When using sheilded wire, a common mistake is to ground both ends of the shield wire. The correct way is to ground only one end. Tying both end to ground can sometimes be worse than no shielded wire at all.
The 20 Ohm resistor from serin to ground is too low. This could cause excessive current in the AXE027. It should be no lower than about 1K, but most use 10K.
The serin current limit resistor is missing from the schematic. While not absoultely necessary when using the AXE027 cable and a 5V Picaxe supply, a 22K resistor is generally recommended, however anything from 1K to 22K will help prevent noise on this line from affecting the Picaxe. I would use a 22K.
The L293D datasheet says that the input current needs to be a minimum of 100 microamps. So to be safe a 500 microamp current will more than cover it. That means that with a 5V supply you could use 10K resistors from the Picaxe to the L293D inputs. 4K7 resistors will allow up about 1ma. I think 330 ohm is too low in this case. The higher the resistor value the less noise that will be fed back into the Picaxe. I would try 10K or 4K7.
Set all unused pins as outputs. If c.6 is not connected tie it to ground.
If it is HV noise on the power supply disabling (constantly resetting?) the Picaxe, then a 10 to 20 Ohm resistor can be placed in series with the 5v supply and the Picaxe +V pin. It should be placed close to the Picaxe. Along with this resistor tie a bidirectional transorb diode to ground. The transorb should be on the Picaxe side of the resistor.
I might suggest an ST BZW04-5V8B. This is a 5.8V TVS diode. A TVS diode could also be placed on the 5V input of each L293D to absorb HV Spikes.
As a test I would program the chip with something like the following.
Code:
#picaxe 20M2
#terminal 4800
dirsC = 255 'all pins are outputs low
dirsB = 255 'all pins are outputs low
sertxd "Starting"
do
pause 100
Loop
Then start the plasma cutter and see if the Picaxe continously resets. If it does, you will see "starting" repeated over and over. Then I would make changes adding dodes, caps, filters, shielding, etc until the resetting problem was solved.
I am not too excited about using the ATX supply for the Picaxe. I might consider taping off of the 12V and then use a "local" 5V LDO regulator near the Picaxe. A 15V TVS didode could then be placed on the input side of the regulator and a 5.6V TVS on the Picaxe side.
A 100uF capacitor might be a good idea on the 5V Regulator output and at least a 10uf cap on the input.
Good Luck