drainplywood
I'm making a shelf above my kitchen sink with intention of putting glasses there to dry off.
How can I treat plywood surface to make it into drainer board?
The sink and the shelf are inside oblique corner, so I can't just buy a dish rack.
Something like 
would work for me, but I need it trapezium-shaped.
I think the best method is to use an anti slip tape / strip / paint
Tape
Strip
Paint
They are ordered form easiest to hardest.
The strips will be the easiest because you just stick them on. You will obviously stick them on the length where your tires will be going and not sideways (because that would be pointless). The strips are more durable, but don't be deceived by the stick on ones once they are stuck.. they are STUCK! The best, but most work is painting the whole ramp.
Preparation.
Needs to be a dry day for sticking on or painting. You need to clean the surface with paper towels - after you wiped the dirt away and the surface you want to stick/paint is dry you will use more paper towels and cheap alcohol (methanol, spirits, etc) - Using the alcohol cleans the surface and removes oil (so don't touch it after you wiped it). Clean it almost like waxing (several times the same spot), this will guarantee your surface is dirt free and guarantees best adhesion for years and years of life.
The strips you can just put down - but you need some cordless drill or elbow grease to get those screws in. Make sure they are flush (level with the strip) or you risk slashing your tires.
Here are some links
http://www.vubasupplies.co.uk/anti-slip-surfaces.html
Laminating a second layer of the same product should stiffen it substantially. Once the first layer is positioned the way you want, spread several beads of construction adhesive across the face of it and conform a second layer over the first. Tack in place with brads using a nailgun to hold it in place until the adhesive sets.
Note: These flexible plywoods come in a variety of thicknesses. It is likely that the thicker ones will be somewhat stiffer than the thin ones. I realize that even the thick ones may not be rigid enough for your application.
Best Answer
Several coats of polyurethane would work. Acrylic likewise. Epoxy would work. Per @DMoore's comment on some other question, redguard would do, but would stink for several weeks (as might the epoxy, depending on epoxy formulation.)
Be sure to coat both top and bottom and especially edges. Top and bottom to reduce warpage that you'd be prone to if you only coat one side, and edges since plywood likes to fall apart if moisture gets in between the layers.
You could also possibly build this from "solid surface countertop" material - 1/2" thick plastic, pretty much.