Freddie here from the Shopify Support team, happy to help.
While I don't have expert coding knowledge myself to be able to provide you with a tutorial on how this can be implemented, Brooklyn is a Shopify Supported theme. This means you can send in an email to email@example.com with this design request, and our Theme Support team can work on your store to implement this feature for you.
With Shopify supported themes, once you are on a paid monthly plan, you get 60 minutes of complimentary design time from our Theme Support team to help make any design changes you'd like to your storefront. You will just need to send in an email to our support with your design request to get in touch with them!
I do want to note for this specific request, while it can be done, it's important to know that having more information required to sign up for a newsletter in addition to just entering in an email address, has a direct correlation to a decreased amount of sign-ups. People don't tend to be comfortable giving extra personal information to businesses they are unfamiliar with or haven't shopped from before. Having said that, if you are confident that your business needs this information, feel free to send us an email so we can help out!
Let me know if this helps, and if you have any further questions.
Shopify Support - firstname.lastname@example.org