JSFiddle - React, Tailwind, and code Playground

HTML

<label for="settings_pass_old"><span>Старый пароль</span></label><br />
                <input type="password" class="settings_input"  id="settings_pass_old"/><br />
                <label for="settings_pass_new"><span>Новый пароль</span></label><br />
                <input type="password" class="settings_input"  id="settings_pass_new"/><br />
               <button type="submit">Сохранить</button>

CSS

.settings_button_active { background:green}
button {background:gray}

JavaScript

$(".settings_input").keyup(function() { 

    return ($("#settings_pass_old").val().length > 0 && $("#settings_pass_new").val().length > 0) ? $("button").addClass("settings_button_active") : $("button").removeClass("settings_button_active");

});