Click Deposits & Withdrawals which can be selected as one of three options of the drop-down menu of Balances, which is at the top right hand corner of Poloniex exchange after a successful login.
Scroll down till you get to DAO (a) as show in picture below and click Deposit (b) to the far right.
Your deposit address(a) in yellow text would become visible as shown below. You have the options of either copying via CTRL + C (keyboard) or Right Click the highlighted (a) with your mouse and select copy. Another option is to deposit via a QR code (b).
MyEtherWallet (browser based client): Follow all the steps as described underneath 5(a) of Kraken token deposit with the only difference being that the To Address in 5 (a) iv will be your deposit address(yellow text) in 3 above.
Mist Official Ethereum Wallet:
Via another exchange e.g Kraken: Follow all the steps as described in Kraken token withdrawal but at 3, you would enter the description Poloniex in (a) and the deposit address you copied in 3. above into (b) and click save address. At 5, select Poloniex Address as your option and continue the process as described.
After a successful transaction, you would see the following screen when you click Deposit & Withdrawal History (c) in 3 above. Initially, it would say pending (It took a while for this to show underneath the deposit history).
After which you would see the screen below after 30+ confirmations:
To view your new account balance, click Deposit & Withdrawals as in 1. above, and scroll to DAO as in 2. above and you should see your new balance as shown below: