So I have an old version of the mist wallet 0.3.7.
How do I update it? I have a few hundred $ worth of coins in there and I don't want to lose them.
I don't see any way to back up the wallet or extract the private keys.
Depending on what OS you have, you just need to navigate to the right folder (below) and copy the entire keystore folder to a new location. I would personally backup this folder on my desktop (temporarily while I update Mist) as well as on a USB drive.
That way, in the future, if it goes missing or your HDD dies, access to your wallet will be lost. Each file inside the keystore represents one account.
You need the keystore file + the password to access each account in the case that your computer / these folders disappear. In my experiences these files have a name like UTC--2016-< MONTH >-< DAY >-...--< ADDRESS >.
Mac: ~/Library/Ethereum/keystore
Linux: ~/.ethereum/keystore
Windows: %APPDATA%/Ethereum/keystore
The above will only back up your normal accounts. Wallet Contracts can be manually re-added, or you can back up the data folder. As far as I can tell:
Mac: ~/Library/Application Support/Mist/"
Linux: ~/.config/Mist or, in earlier versions: ~/.config/Chromium/Mist (folder is hidden)
Windows: C:\Users\< Your Username >\AppData\Roaming
The folders I see online tend to be all over the place. I can confirm that the above certainly hold true to my Mac.
To update, simply visit the ethereum wallet releases page, download the zip for your OS, and run the install. In theory (and with my experience) this doesn't touch your keystores. But, better to be safe than sorry.
Regarding your security conserns: How to backup mist wallets?
Usually your keys are untouched when upgrading. Go to the downloads page and grep the latest release, currently 0.5.1:
And verify the sha sums:
SHASUM Ethereum-Wallet-linux32-0-5-1.zip: 896b8fbbb24d67a42e6ce4c4cf96de764af6be34
SHASUM Ethereum-Wallet-linux64-0-5-1.zip: 1b90a93ac508511e76c533e6ce76c89919861a1b
SHASUM Ethereum-Wallet-macosx-0-5-1.zip: b523052fe2527878ece408bf86805840741a6b55
SHASUM Ethereum-Wallet-win64-0-5-1.zip: 680fc51669dbcb63d351b5bd5d2f332406cfe876