Why not use Sting's light?
 

When the Company left the Chamber of Mazarbul, leaving Gandalf at the top of the stairs leading away from the Chamber to try and lock the door, it was stated that the Company was in great danger because in the darkness they couldn't tell when flights of stairs started and they might fall. But surely with the close proximity of Orcs if they had taken Frodo's blade Sting it would have given off enough light to guide them?

Unreliable. It would grow dimmer the further one moves away from Orcs. Besides, they had a more reliable light source, Gandalf's staff:

You've found a plot hole! Bad writing! The Internet!

But more seriously, no, it actually doesn't give off sufficient light. This is consistent with the original description in The Hobbit, where the party also has to rely on Gandalf's light to navigate the Goblin caves, despite also having both Orcrist and Glamdring, and where Glamdring is quite obviously described as glowing enough to see the sword by, but not enough to cast light on it's surroundings.

True, Bilbo is subsequently described as using Sting but he's also described as feeling his way around a lot, and that was in narrow tunnels rather than a great hall.

I would just add that the purpose for Sting (and Glamdring/Orcrist) glowing was not to provide light; it was to alert the user of the presence of Orcs. It would even seem that a very strong light could be a liability when enemies are about.

When the Company was descending the stairway from the Chamber of Mazarbul Gandalf remained at the top of the stairs, so Glamdring's light or illumination from his staff were not available.

Also at Parth Galen Aragorn asked Frodo to draw Sting to see if orcs were nearby, and although at that moment Sting was faint it did show that orcs didn't have to be in close proximity for a light to be generated.