Binance has confirmed support of the token registration for it's users, so for that reason you can leave them on the exchange safely and still be credited with the official EOS coin when it launches.
However just because they will auto-register you doesn't mean you will also receive all the possible airdrops that will be happening in early June. The safest course of action to maximize your airdrop participation (and revenue) is to hold your own EOS in a wallet (off-exchange) that you control the private key for - due to this, you should register your own tokens and not rely on Binance.
As for trading, the ERC20 tokens will be permanently frozen so you won't be able to trade between June 2nd and the time the mainnet launches and you're credited your own EOS Coin.
For all users looking into registering your tokens, I'd recommend doing so immediately as the first airdrops to require registration are starting in the next 36hours (May 10th 2018). https://steemit.com/eos/@evolutionos/evolution-airdrop-may-10th
Here's a very useful guide on how to register your tokens offline for maximum security created by EOS New York (a bp candidate). The guide uses the official EOS repositories so there's no need to rely on any 3rd party code.