2 problems with climbers, or 3 really: first, climbers want to climb up, not sideways; second, they never reach their full potential in terms of height and spread in pots; third, the railings dictate you'd have to use twiners and not clematis, for instance (although its hard to tell how thick the railings actually are). That said, you could try Passiflora caerulea in a pot 2 feet deep by 12 inches wide - with a bit of rigorous and frequent training and tying in, you might persuade it to go for a way along the railings rather than just up in the air. Maybe a wisteria, trained along the railings too, although it won't get very big contained in a pot. Clematis montana might work for a while, provided you tie it in to the railings - that one eventually twines a bit. The drawback is obvious though - a pot 2 feet deep is nearly at the top of the railings already, but without that depth, climbers won't do.
I wouldn't use climbers though - I'd use the same size pots as mentioned previously, and put things like different bamboos (phyllostachys nigrescens, for instance) and Phormiums (although that depends where in the country you are). That doesn't look like a very big area, so you probably don't want wide pots there, take up too much space, so anything that prefers a deep but narrow pot would be my choice.
I have a balcony with railings like this - I use troughs at the front which I plant with summer bedding to trail over the edge, interspersed with tall, narrow pots containing various phormiums. I do have two climbers in deep pots, both clematis, one with an obelisk support in the pot, and the other growing up clematis mesh attached to a small wall leading up to the roof overhang, so nothing growing on the railings themselves.

Thanks for your advice - really helpful! Going to research your suggestions.

Are you going to try to cut down the wind? I create shelter in the vital part of my garden using horticultural wind-break netting. It is also equally called 'shade mesh'. I attach it to trellis.