And, all the parts that Brent mentioned you can find, packaged and ready to use, inside a solid state switch. It has 4 terminals, two large ones for the switching and two smaller, optical isolated ones for the control.
If you are talking about a simple stick welder you could try the following, if yours is a MIG welder, please disregard as I have not tinkered with these so far. For welding thin metal TIG is the way to go I'm told - wish I had one of these.
Now that you are telling us what you are trying to do, I think fast switching is perhaps not the best way to reduce the current with a welding transformer. It may cause complications to the welding process.
I take it there are some current settings on your transformer (most welding trannies have them) but none are low enough for your requirement. Some transformers regulate the current by moving part of the core, perhaps there is room for extracting it further if yours is that type.
To get a lower secondary current you can also lower the primary voltage to the transformer. It would, however, also lower the 'striking' voltage somewhat.
You need to connect something in series with the primary that causes a voltage drop. Have a look inside and see if there are several taps for different primary voltages. If so, just connect it to the next *higher* tap which, ideally, is no more than 20V higher than your supply voltage.
For example, I did the reverse on my welding transformer which had a 125A output that was not quite enough. So I reconnected the primary from the set 240V to the 220V tab and this solved the problem. Yes, feeding 240V into a 220V winding does make things hotter in there but since my welding was only for short bursts (stick) there was no heat problem.
Another thing you could try, connect the transformer via a long extension lead (uncoiled!) which has a low ampere rating. The voltage drop from this might be enough to lower your secondary current sufficiently.
I would not mess with the high current secondary output, switching that kind of current is just too troublesome. Mechanical relays would just weld the contacts together and solid state could be expensive.
IF you do try the above tricks, keep in mind you are dealing with lethal mains voltages and take the appropriate care.
Klaus