The hanging baskets traditionally used for orchids are made out of teak or redwood slats, even thick metal wire, but any kind will work. You just want to avoid the kind that have a ton of sphagnum or that stringy stuff (forgot the name of it) inside because it'll hold too much moisture and end up rotting the roots.
Water the plant first to make the roots more pliable, then squeeze the sides of the pot to loosen the plant. You may want to wait to repot it until after it's done blooming: Brassia roots are pretty thin and easily damaged.
When you get the plant out, very carefully pull the loose sphagnum it's potted in away from the base and roots. If the roots have grown into the stuff to the degree that you can't remove the sphagnum easily, don't damage the root system: better to leave it alone and remember not to overwater.
Here, we anchor our hanging plants (usually Vandas) in baskets using lava rock and tuck the longer roots between the slats so that they hang out. In your case, the roots will be nowhere near the size of a Vanda's, so it's just a matter of filling the basket with lava rock or coarse redwood bark to the level that you need to raise the plant up to. Place the plant in the basket, then anchor it down with more coarse bark or lava rock, just enough so that it stays put.
And that's it! Not very complicated. As the plant grows, its roots will extend and "grab on" to whatever you used to anchor it with and it will steady itself naturally.
You can buy lava rock at any home and garden center: the kind used for walkways works great. Just rinse it to get the dust off first, and you're good to go. The size or grade you use will determine how much water is retained: the finer the grade, the more water will be held. If the basket you're using has very wide openings, you may have to line the inside of it with chicken wire mesh first, and then go ahead and fill it with the bark or lava rock.