The web wallet doesn't not work well with Safari .. 
As far as I know this has to do either with the websocket implementation or the local database store ..
We had a warning for those browsers earlier .. not sure what it has been removed.
Could you try Safrai on a Mac and open the development console to see if there are javascript errors?
Thanks 
@xeroc.   I wish I had a Mac to test on.  
I don't think it's entirely a Safari issue because from what I understand the iPhone6/Safari works fine.
As you mention it may have something to do with websockets and particularly on ios7 devices.   It doesn't initialize correctly.  The url always goes to: 
https://bitshares.openledger.info/#/init-errorHere are some links to resolving similar Safari/iOS websocket  issues. Hope it can help:
http://stackoverflow.com/questions/5574385/websockets-on-ios?answertab=votes#tab-topMatthew Levine's fix: 
me = this // strange javascript convention
this.socket = new WebSocket(url);
// put onmessage function in setTimeout to get around ios websocket crash
this.socket.onmessage = function(evt) { setTimeout(function() {me.onMessageHandler(evt);}, 0); };
Mobile Safari crash when returning to WebSocket page
https://gist.github.com/mloughran/2052006Safari (IOS and osx) crashes when connected to websockets
https://github.com/SignalR/SignalR/issues/2650