Go to / Scroll to Password Field on WordPress Posts After Submit With Message(s)

Not exactly sure what you are trying to do and this doesn’t look like a WordPress issue, but submitting a html form will certainly lead to a new page load. So what you want is redirect not to the same page (which happens when you specify no action, leading the page to scroll to the top) but to an anchor on that page. The usual way to do this is to include it in the action variable of your form like this:

<a name="somewhere"></a>

<form method="POST" action="#somewhere">
  ...
</form>