The main danger when using MEW is a chance of providing your private key to a malicious site after which you immediately lose all your ETH and tokens. To avoid this situation you should always check whether a connection is secure by looking at the address bar.

Here's an example example of a phishing site were connection is not secure

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I have easily found it by googling "mi ether wallet". That's what they do, they exploit your mistakes, so please be very cautious.

Also, apart from looking for a secure connection you have to double check whether the address is looking exactly like this:

https://www.myetherwallet.com/

Btw, I just checked, the phishing site in my example is the top one you see even when googling the write words: "my ether wallet" because that's a paid Ad. Thanks a bunch, Google!