Can I utilize unused 30amp circuit for 20amp welder?
TL;DR Replace the double 30A breakers with double 20A breakers. Keep the 10 AWG wire. Replace the dryer receptacle with 2 20A receptacles wired as a MWBC - one hot to each, neutral shared. No safety notes or special procedures once the changes are complete.
And now all the gory details:

Replace the double 30A breakers with double 20A breakers


You can't have devices designed for use with 15A/20A circuits on a 30A feed. There are certain very specific exceptions (e.g., a 15A receptacle that comes pre-installed as part of an electric range, where the range is itself connected to a 30A or larger circuit), but in almost all situations, if you are going to either install 15A/20A receptacles (whether 120V or 240V) or hardwire devices that are designed for 15A/20A circuits, you must replace the 30A breaker with a 20A (or 15A, but that gains you nothing in this case, so go with 20A) breaker.

Simply removing one of the 30A breakers doesn't help. It does remove power from one of the hot wires. But it does nothing to change the 30A vs. 20A problem.

Replacing both 30A breakers (and they are really a pair, removing just one would not be so easy anyway) with a double 20A breaker provides advantages for use as an MWBC (Multi Wire Branch Circuit) which (a) lets you get 2 20A 120V circuits without changing the wires and (b) lets you use 20A 240V equipment as well.



Keep the 10 AWG wire

This may seem obvious, but it includes some important details. You can use 10 AWG wire with 20A breakers (should be no problem at all) and receptacles (if they can't take 10 AWG wire directly, you can connect using a pigtail). Leaving the 10 AWG wire in place avoids having to run new wire now (which may be trivial or may be a lot of work) and also allows future usage for a 30A circuit if desired.

Replace the dryer receptacle with 2 20A receptacles wired as a MWBC - one hot to each, neutral shared.

This is where it gets interesting. The existing dryer receptacle, if original (or straight replacement) to the 1959 build, is a 3-wire connection. This includes two hots and a neutral, but not a separate ground wire. This has safety implications, which are actually easier to resolve with 20A 120V circuits than with a 30A 240V circuit! The methods include:

If you have metal conduit or an existing (but unused because of the 3-wire connector) ground wire, use that as the ground for your new receptacles.
If you have an accessible ground from another circuit 20A or larger (so that the ground will be large enough for this circuit), you can run a ground wire from the new receptacle to that circuit's ground.
You can install a 20A GFCI and label it &quot;No Equipment Ground&quot;. This gets tricky (but doable) with MWBC (either only use 1/2 the MWBC, or install two separate GFCI receptacles).

Any of these solutions will take care of &quot;old dangerous dryer circuit without ground&quot;. Note that if the receptacle is in a location that requires GFCI (which is a lot of places now, depending on your state's version of the NEC), you may be required to install GFCI, either at the receptacles or using a double 20A GFCI breaker.
As far as MWBC, this lets you use any combination of:

240V circuit - black hot/red hot
120V circuit &quot;1&quot; - black hot/white neutral
120V circuit &quot;2&quot; - red hot/white neutral

which conveniently lets you get two 20A 120V circuits without running any new wires.

No safety notes or special procedures once the changes are complete.

This is crucial. For some oddball separate device, you can get away with your own funny procedures. But you can't do that for hardwired equipment or receptacles, because procedures will be forgotten. For example:
You leave the 30A breaker in place and wire up a standard duplex 120V 20A receptacle. You sell the house in a hurry, forget to (or run out of time to) change things back. New owner has no idea. Plugs in 2 or more 120V devices at the same time rated for high power - e.g., a couple of space heaters and some tools. Total power usage goes to 40A. Not enough to trip the 30A breaker for quite a while. But enough to overheat the 20A-rated wire (especially if it is older cable).
A proper installation - with the most important piece being the 20A breaker - avoids any special requirements. What you or any future owner can use is determined at a glance by the receptacles available. Unambiguous. Safe.