The guards inside are, strangely enough, on Dreadguard Inner faction.
[You must be logged in to view images. Log in or Register.] This faction is raised through the
Red Wine quest and/or the
Evil Research quest. The quickest way will be the Red Win quest, just got some invis pots, buy tons of Red Wine from a friendly vendor somewhere, and you'll have good faction in no time.
Be careful about two things. First of all, on your way to the Red Wine NPC in 3rd quarter, there's a very KoS Cleric guild - hence the invis pots.
Second, once you have faction with the Inner guards: Just outside the entrance to the bank in Neriak is a level 40 Wizard who warps down from the caster building and patrols around kinda aimlessly just in front of the door to the basement leading to the bank (it's a bug I reckon, but it's like that on Live EQ too, so it is proper behavior for our server). He's threatening to me (Iksar Monk), so he doesn't agro from that far away, but you gotta tippy-toe around the corner as you go through the merchant buildings, and see if he's down on the "street" or up on the caster building. Feign Death, watch him patrol, and move in when it's clear.