Multiple string matches with indexOf()

'Video' || 'Audio' is a logical OR. A non-empty string is implicitly a true value in JavaScript, and thus the short-circuited OR is not evaluated and this collapses to just 'Video'. This is why you see the results you do.

Others have pointed you in correct directions to resolve.

It's far shorter to turn this into a regular expression.

if ( srcTag.match( /(video|audio)/ ) ) {
  /* Found */
} else {
  /* Not Found */
}

On a side note, please don't do what you're attempting to do. Asking users to download Safari when they're using Internet Explorer 8 is doing a disservice to the Internet, as well as to that user.

As for redirecting the domain to another location, you should use .preventDefault() to keep the browser from following the link:

$("a.videoDownload").on("click", function(e){
  e.preventDefault();
  if ( this.getElementsByTagName("img")[0].src.match( /(video|audo)/ ) ) {
    window.location = confirm( 'Download Safari?' )
      ? "http://apple.com/safari/download" 
      : "../../../index.html" ;
  } else {
    /* No match */
  }
});

Again, please don't actually do this. Nobody wants to be that guy, and when you tell users to download another browser, you're being that guy.

I think you probably want the OR (||) operator outside indexOf as so:

if ((srcTag.indexOf('Video') !== -1) || (srcTag.indexOf('Audio') !== -1)) {
  ...
}