Question: How did the League of Shadows use economics to attack Gotham?
Answer: They used their influence to trigger the economic depression that was gripping Gotham when Bruce was a child.
What type of influence did they have?
It's never specified in the film, so any answer would be pure speculation. They merely say they attacked Gotham economically in the past. I'd presume they'd use power and threats to do things like tank companies, make people lose their jobs, increase homelessness, make it more difficult for people to get help, etc. Basically, just ruin the citizens financially.
Continuity mistake: As Ducard is sliding down the ice toward the cliff, he's on his front, sliding feet first. Bruce is also on his stomach, but sliding head first. Then, just before Ducard goes over the edge, Bruce is instantly on his back, sliding sideways, right side toward Ducard, just in time to take Ducard's hand, saving him. (00:40:10)
Suggested correction: The shot from afar shows Bruce rolling as he grabs Ducard at the edge. He isn't instantly on his back.
The mistake is valid. In the shot from afar, Bruce is already on his back with no time to roll over into that position from the previous shot.
Answer: By having their own people working inside the trading company, they would do a pump and dump. Take a small company, put the word out that it's the next big thing, watch the prices rise, then sell. Like insider trading, the millionaires become billionaires, while the billionaires become broke.