That's a fairly common way to design a form both on paper and on the web.
I'm not quite sure exactly what you mean with a better way to do so...
If you're worried about the hidden field not appearing if the user has javascript disabled, I'll suggest you hide the field using javascript or have a duplicate "If other please specify" text area in a noscript
block:
<select><!-- implemented something like rahul showed -->
<noscript>
<label for="ifOtherInput">If other please specify</label>
<input type="text" name="ifOtherInput" id="ifOtherInput">
</noscript>
<!-- This is initially hidden and shown by when the user selects the other option -->
<div id="divOther" class="dispnone">
<!-- Here we know the user selected other so we can just have this label: -->
<label for="ifOtherInputJs">Please specify</label>
<input type="text" name="ifOtherInputJs" id="ifOtherInputJs">
</div>
The backend must handle that the input in the noscript block may be missing. Or you could add the javascript version of the input to the page of the input using javascript (so both cannot possibly appear simultaniously so that they can have the same name.