You can take a look at these options. Do you have dojoConfig before the ArcGIS JS API script tag:
<!-- set Dojo configuration, load Dojo -->
<script>
dojoConfig = {
parseOnLoad: true,
async: true
};
</script>
This may provide you with a better error message when debugging.
If it's just on the client machines I would think it's the IE settings like Kelly Hutchins mentioned.
It can be tough to figure out where to start but I would start with security settings.
Click on Tools >> Internet Options
Select the Security tab and then select Custom level button; this will open additional settings where you can see what may be blocked, enabled or prompted. You should compare this to a machine that has a working application. There are four areas to check here; Internet, Local intranet, Trusted Sites, and Restricted Sites. Only change one item at a time if something doesn't match, click apply and refresh the page. If the security settings match, you can compare the other options one by one.
Also, ensure any security setting that you change actually stayed. You can close the browser and re-open it for this. If they don't, there may be security preventing browser security changes.
