By default bbPress (WordPress forum) has the notification checkbox UNchecked.
To make this default checked, create a folder called "bbpress" in your theme folder.
Copy from your bbpress plugin folder
/wp-content/plugins/bbpress/templates/default/bbpress
the following files into the newly created "bbpress" folder of your theme:
form-reply.php
form-topic.php
Locate in both files the line
<input name="bbp_topic_subscription" id="bbp_topic_subscription" type="checkbox" value="bbp_subscribe"<?php bbp_form_topic_subscribed(); ?> tabindex="<?php bbp_tab_index(); ?>" />
and change them to
<input name="bbp_topic_subscription" id="bbp_topic_subscription" type="checkbox" value="bbp_subscribe" checked tabindex="<?php bbp_tab_index(); ?>" />
Notice that added "checked" after the value field and the removal of the PHP part with bbp_form_topic_subscribed() in it.
Save both files and as of now the checkbox will by default be checked.