How to check in javascript if web part zones have no web parts?

You can check if a page has some webparts or not.

var siteUrl = '/sites/MySiteCollection';
var serverRelativeUrl = '/sites/MySiteCollection/Default.aspx';

function updateWebPartTitle() {    
    this.clientContext = new SP.ClientContext(siteUrl);
    var oFile = clientContext.get_web().getFileByServerRelativeUrl(serverRelativeUrl);    
    var limitedWebPartManager = oFile.getLimitedWebPartManager(SP.WebParts.PersonalizationScope.shared);
    this.collWebPart = limitedWebPartManager.get_webParts();    
    clientContext.load(collWebPart);    
    clientContext.executeQueryAsync(Function.createDelegate(this, this.SuccessFunc), Function.createDelegate(this, this.onQueryFailed));
}

function SuccessFunc() {    
    if (!collWebPart.get_count()) {
        alert('No Web Parts on this page.');
    }         
}    

function onQueryFailed(sender, args) {    
    alert('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace());
}

Check MSDN source here.

Update:

If you know the webpartzone id and try something like this.

<div class="inner"> 
<WebPartPages:WebPartZone id="zone1"><ZoneTemplate></ZoneTemplate></WebpartPages:WebPartZone> 
</div>

then query the zone and see it has table element because generally webpart will be rendered in a table.

if($('#zone1').find('table').length != 0){
        //webpart exists
}

You may need to adjust your code based on actual dom.I can not test it now.