Need advice about rewiring detached garage
That thing is more of a shutoff switch -- the fuses just happen to be "a thing" they often put in shutoff switches.  A shutoff switch is required in an outbuilding, and so for that reason, I would retain the switch even if you changed the wire.  It looks like a fine switch to me. 

GFCI and done

This uses GFCI to provide electrical safety on the existing setup. 

The simplest way to provision the power you want is to install a junction box, wire the cable out of the shutoff switch so its first stop is this junction box (which may already exist) -- and install a $20 GFCI+receptacle combo device. Voila. 

GFCIs only protect stuff downstream of them, so it's best to put the GFCI protection in the house, either as a breaker in the main panel, or a GFCI device (+receptacle or deadface) enroute.  

Subpanel

With this option, you must trench a separate ground wire back to the house.  

You cannot drive a ground rod and call it done, for reasons often discussed here. Given that 90% of the cost is in the trenching, you almost might as well just run new and proper /3+ground cable of the ampacity you desire. 

A subpanel is almost a lost cause because with only 20A (12AWG) or 30A (10AWG) capacity at 120V, you won't be able to support more than 2 circuits and no more than 2400/3600W.   You cannot support 120V and 240V loads using this method. 

I would retain the shutoff switch because there's nothing wrong with it.  But if you use a "main breaker" type subpanel, that "main breaker" satisfies the code requirement for a shutoff switch.

Keep in mind when you run a common subpanel off 120V only, every other row in the subpanel is dead, so you'll use the spaces at twice the expected rate.  Yet another reason to get an ample sized subpanel. 

DO NOT consider installing a generator interlock panel on the hopes of backfeeding the house using this cable.  Ask us how to do that if you want it.   You'll still have to trench something though. 

New main cable

In this case you trench and run a 12/3 or larger + ground cable.  Rearrange the shutoff switch so the switch interrupts the two hot wires, and the neutral bypasses it.   From there you either


run it as a multi-wire branch circuit, splitting each hot into two circuits along with the shared neutral and ground, breakering it for 20A.  
run it into a subpanel as above, but using both hot wires and separating ground from neutral.  You still need a local grounding rod. 


Transformer; new main service.

No ground wire retrofit needed, but it requires a transformer.  It will allow you to exceed 3600W in the garage. 

In this scenario, you come directly out of the legacy shutoff switch straight into a transformer's 240V primary.   The transformer 120V/240V secondary goes to a new sub^H^H^H main panel. Since it is a main panel off a transformer-isolated service, it must have a local grounding rod and must bond neutral to it.  Running a ground back to the main building would be unhelpful. 

Back in the house, you punch these two wires down to a 240V breaker of correct size to protect both the wire and the transformer.