Enter the bank, at the corner of Brite St. and Lansdale Ave. Explore the main hall of the bank. There is a flashlight near the window in case you need one. When you are done exploring the main hall, go all the way back and through a double door. There are some offices in the back, you should find a first aid kit in an area near some cabinets, and also a crowbar on the floor.

Locate the stairs, and climb up to the second floor. Enter the Bank Manager's office. There is a fire axe near the wall, which makes a good melee weapon, so you should take it.

Go back down the stairs, and then go down one more story. There is a safe vault there. Enter the vault. There are five safety deposit boxes that can be opened. Open the first safety box. This sounds the alarm,  and also gives you a side quest, "Bank Mystery".

Silent Hill Downpour Bank Side Quest

Bank Mystery
Discover why the bank alarm keeps sounding.

In this mission, you will be facing five waves of increasingly hard monsters.

Go to the ground floor. There will be a Screamer there. Kill it, and the alarm will stop, and the first safe deposit box will open. Go back to the vault, and search the deposit box to find a First Aid Kit.

Also, around this point, you might have already killed 10 Screamers, giving you the following achievement/trophy:

Silence is Golden
Killed or incapacitated 10 Screamers

The alarm keeps sounding after you have emptied the first deposit box. Go back to the ground floor, and this time find two screamers there. Kill them, and a second safe deposit box opens. This one contains a box of pistol bullets.

Kill Switch: If you go to the second floor while the alarm is sounding, and you enter the bank manager's office, under his desk there will be a red blinking light.  This is a kill switch to shut down the alarm, but you do NOT want to press it. Pressing it will shut the alarm but also close the vault, forcing you to leave the bank and re-enter.

Since the alarm is still sounding, go back to the ground floor, this time find a Screamer and Brawler there. After you neutralize them, a third safe deposit box opens. This box has a Colt handgun.

The next wave of monsters includes 2 brawlers. Once you kill them, a fourth deposit box, containing some more pistol bullets, opens.

The last wave will include a brawler, screamer, and stabber. Defeat them to unlock the fifth (and last) safe deposit box, which contains three first aid kits. You also unlock the following achievement/trophy:

Silent Alarm
Completed The Bank Side Quest