Q. How does luck affect drops? Does Lady Luck help? A. Get fukt nerd hahahahahahaha  -Donny
Q. How could i increase my luck? A. “Sleep with the fishes”  -Slim
“I saw, I came, I came” -Warrior
"I didn't type this! ^" -The Real Warrior
"You're lucky if it works" -Phoenixf
"Jump exactly 10 times before opening a cache for extra luck" -Utjan
“Luck whats that? and how do i steal it from people?” -Nosjo
“You know what they say, 3rd times the charm.” -Xehno
“Luck is when you don't know how you succeeded” -Dreggy
“Luck has a maximum, if you go over it it goes back down” -MoonLight

The Real Info

Only not getting rare items at all(increases) modifies your luck now, any previous accessory luck modifiers have been changed and their luck bonuses added to the base stat!

If you are the exceptionally unlucky kinda guy, it helps as the longer you active chance for those (like beating bosses, opening caches, citadel etc) - the chance gets easier until you get something like a weapon mod or higher! The scales prevail!