I cant say sitting here if a 50 amp breaker will pass whatever current the welder draws without knowing the welders primary current draw at the heat range youre welding at, but it is likely if youre welding at a low amperage setting it may well do so. What matters electrically is if the branch circuit wiring has sufficient ampacity. If you want to weld at higher amperage (that would trip a 50 amp breaker) then you have to increase the branch circuit wiring ampacity and then protect it with an appropriate sized breaker.
The way its done (at least how I did when I was a design engineer) is you FIRST calculate the load,,,,,,,Then you select feed wires having an ampacity of at least 125% of the maximum continuous load,,,,,,,,Then you select the overcurrent protection device appropriate for the wires ampacity. For example, on a branch circuit if I calculated the max continuous load was say 15 amps, Id specify 20 amp rated wire and protect it with a 20 amp circuit breaker.
Many typical tombstone AC Buzz Box Welders are straight 240 volt (i.e. NO 120 volt loads and no Neutral required) and the branch circuit feed consists of two Hots (L1 & L2) and an Equipment GroundING Conductor. Of course, if a welder required BOTH 120 and 240 volt supply, then the branch circuit consists of two Hots, a Neutral (GrounDED Conductor) and an Equipment GroundING Conductor.