As far as i know and based on my research throughout the forum, keeping a huge amount of cryptocurrency on a web application is not a good thing in security (and some other) aspect.
However, the accessibility and features that a web app wallet given to you is considerable.

Although, I've never used MEW before, but I use MetaMask for my ETH wallet, and as far as my experience right now, I could say it's pretty safe. Although, I'm considering to move into desktop wallet, something like Electrum but for ETH.

Since you asked if is it safe to keep big amount in a Web Wallet App, well it depends on the app itself, but if you really want that extra oomph of security, I suggest you to use desktop wallet instead.