Edit in JSFiddle

var download = function () {
  // ダウンロードしたいコンテンツ、MIMEType、ファイル名
  var content  = 'abc';
  var mimeType = 'text/plain';
  var name     = 'test.txt';

	// BOMは文字化け対策
	var bom  = new Uint8Array([0xEF, 0xBB, 0xBF]);
  var blob = new Blob([bom, content], {type : mimeType});

  var a = document.createElement('a');
  a.download = name;
  a.target   = '_blank';

  if (window.navigator.msSaveBlob) {
    // for IE
    window.navigator.msSaveBlob(blob, name)
  }
  else if (window.URL && window.URL.createObjectURL) {
    // for Firefox
    a.href = window.URL.createObjectURL(blob);
    document.body.appendChild(a);
    a.click();
    document.body.removeChild(a);
  }
  else if (window.webkitURL && window.webkitURL.createObject) {
    // for Chrome
    a.href = window.webkitURL.createObjectURL(blob);
    a.click();
  }
  else {
    // for Safari
    window.open('data:' + mimeType + ';base64,' + window.Base64.encode(content), '_blank');
  }
}

// Click event
document.getElementById('download').addEventListener('click', download);

<div id="download">Click to download</div>

#download {
  display         : block;
  cursor          : pointer;
  background-color: #6DB7F7;
  width           : 180px;
  line-height     : 2em;
  text-align      : center;
}