To have a product display a "NEW" banner or label, create a new product collection simply called "New" from the Collections section of the Shopify Admin. You'll also want to ensure that the collection's handle text is also "new".
Any products added to that collection will automatically display a New banner:
To confirm or change the collection's handle, you will need to click 'Edit Website SEO' found near the bottom of the page, which will then open the options shown in the image below: