Docs: "Search"
https://www.jamroom.net/the-jamroom-network/documentation/modules/950/search#jrsearch-form
Use the template="" instead.
So
{jrSearch_form template="my_search_template.tpl"}
Then in your skin, add the template file. ( call it whatever you like )
/skins/( YOUR SKIN )/my_search_template.tpl
copy the default template from modules/jrSearch/templates/html_search_form.tpl which looks like this:
{* Standard HTML search form *}
{jrCore_lang module="jrSearch" id="7" default="Search" assign="st"}
{assign var="form_name" value="jrSearch"}
<div style="white-space:nowrap">
<form action="{$jamroom_url}/search/results/{$jrSearch.module}/{$jrSearch.page}/{$jrSearch.pagebreak}" method="{$jrSearch.method}" style="margin-bottom:0">
<input id="search_input" type="text" name="search_string" style="{$jrSearch.style}" class="{$jrSearch.class}" placeholder="{$jrSearch.value|jrCore_entity_string}"> <input type="submit" class="form_button" value="{$st}">
</form>
</div>
into your my_search_template.tpl file, then you can adjust things directly.