Wait.....I can store guns for my next character????? Fook. I just been dropping or selling everything. Never mind, guess that would make my second run too easy anyway.
The 'bank' is in Crimson Raider HQ, upstairs. You use Eridium to buy more room.
As soon as a purple no longer cuts it, they're good for 6 or so levels to me, put it in there.
Claptrap has the 'twink stash' near his little hovel in Sanctuary. You'd move the lowest level purples in there.
When your new character arrives at Sanctuary, he should get to that stash and loot the goods. When he's done with them later in the game, he should put them in his own bank. Repeat this process over and over for the kind of cackling, chortling 'they don't stand a chance' goodness for later characters.
Yes, it may make things too easy, but you play alternative characters for variety more than challenge. The first character gets all the challenge.