I'd first add a front and rear anti-sway bars (or camber compensator if rear is swing axle like most VS Speedsters). For some reason VS doesn't seem to use a front anti-sway bar at all (not even the skinny stock VW one). Plus they typically only use one lowering adjuster in the front beam. Then I'd get a 4 (2 pairs) of caster shims and 2 of the longer bolts for the front beam. Then set tire pressure to whatever is in the VS owners manual (22 psi?) and get an alignment. If you still have front drums swap them out to discs before the alignments. The VW steering box can be tighted up a bit (there is an adjustment nut on top) but they all seem to have play in them.
Not really handling but driver feel - is to replace the stock shifter with a CSP or Vintage shifter. There are others but they greatly reduce slop in shifting.
This is a good book for $10 or so - How To Modify Volkswagen Beetle Chassis, Suspension & Brakes by James Hale
Yes, the steering box has a slotted screw and a jam nut for adjustment, but that is NOT the first thing to mess with! The big nut on the front of the box needs to be adjusted. Most people skip this step because they don't have the proper tools. Don't do it!
You'll prematurely wear your steering box if you skip the large worm adjuster. You need a 41 mm open-end or box wrench, yes 41mm. You also need a 24mm hex key to turn the adjuster. I got the 41mm wrench on Amazon for $18. I welded a 15/16"(VERY close to 24mm) nut to a piece of flat steel to turn the adjuster, it doesn't need much torque to adjust....
I've seen people use an old VW oil pressure switch and cut off the plastic and the threaded end, then weld that to a piece of steel.
As Greg said, get some caster shims, but first check ball joints, tie rod ends, beam bushings, ans make sure the pitman arm is REALLY tight on the steering box. Also check steering shaft rag joint, these do fail. Tire pressure and alignment! And don't forget to check the steering stabilizer.
I'd also add to Ed's comment about the steering box: I have a brand new TRW box in my Spyder. After 1000 miles, it was sloppy, about 1.25" inches of play in the center. After I adjusted both the HUGE nut on front and the slotted screw on top it's better. I have almost zero play at the wheel and MUCH improved confidence in where the car is going!