Is this standard work on your VS? This one is from 2007, and appears to be a drain of some sort...
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We "OLDE TIMERS" know it to be a dump hose for the crank case breather...... Pre PCV valve and whatever is now used..... The other end probably connects to the oil filler neck... Most replicas route them to an air cleaner or breather box..... 
As Leon said, it's the drip tube for the breather. I can't see your pic full size; it's looks like they've taken the breather hose off the oil filler breather fitting and just run it down past the bottom of the engine? As well as letting the engine breathe out it can also pull dust from the road in. It would be better hooked up to a breather box (as Leon also mentioned). Al
Ditto, what ALB and Leon said.
Factory VW dump hose has a rubber cap on the end with a small slit rather than a full 1/2" ID hose opening as seen.
The MGB had an interestingly named (and shaped) drain hose coming out of the vent just before the windshield.
Looks like air deflector pieces were left off or removed too?
...who needs 'em...what does the factory (VW) know about cooling?
(tongue securely in cheek)
"Hans & Fritz " are spinning in his graves....
Alan Merklin posted:Factory VW dump hose has a rubber cap on the end with a small slit rather than a full 1/2" ID hose opening as seen.
Which is to allow the air passing over it create a vacuum in the tube to pull the vapors out while also making it difficult for dirt and debris to go back up the tube.
If using a completely open tube, then it should be cut at an angle so the longer side is toward the front of the car. This way it creates low pressure on the back side of the tube to pull the vapors out. Do it backwards and you've created an air scoop to blow crap into your engine.
You guys always have such great advice/info.
So, am I the only VS owner with no "air deflectors." Is that standard issue on all of them? Cant seem to find anything else about it in these forums, either. They appear fairly easy to put on?? Any sourcing advice for these, or links to more information about them.
Any advice about a more permanent and clean solution to the dump hose would be great too. How often are breather box's used in these builds? Ive seen some pretty creative installs on the forum here.
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Here's a pair -C & D $5 ea
I've never seen any tin work pieces like that on mine. So I don't have any either. If I did I wouldn't know were they are suppose to go! Oh well.
I have a breather box, no dump hose, although my more than mild engine, is probably better suited to a breather hose, off of the oil filler.
Art
they go on the back of the motor (the back of the car) right above the exhaust manifold flange on each side. The smaller of the two goes on the right and has an accommodation for the stock Oil Filler / Breather dump nozzle. The larger (the top one) goes on the left.
They are fastened from above through the cylinder head tins with those famous screws we all know and love.
When people say "You have to have all the stock tins..." - these are the ones I think of.
And the sleds (all 3) and the heater box delete pieces as well...
OCAP1 - FWIW, I had the same breather tube set-up as you have (old school VW crankcase breather) on my previous daily driver VS (1835cc).
Over 15 years of driving it all over the Western U.S and logging 100,000 relatively trouble-free miles, my engine never had issues with this.
I have the same set-up on my current daily driver VS (1915cc). 2 years and 25,000 miles later, guess what, no problem. 
I'm of the keep-it-simple school of thought.
"Welcome to The Madness" known as replica Speedster ownership. 
Here's wishing you many miles of top-down driving enjoyment with your VS.
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Don't get me started on "Sled Tins"........
And having a breather tube that outputs onto the ground is just fine.