Welcome to the Shopify Community! Thanks for sharing a screenshot of how your Brooklyn theme is currently appearing. From looking at the image, I'd understand why you'd want to make this space smaller.
However, the design of the Brooklyn theme isn't a small header / banner like you see in other stores, or themes. The home page of the Brooklyn store is intentionally one image that takes up the whole space above the fold (before scrolling down). We can see a great example of this design in the Brooklyn preview store. The image takes up the entire home page, until you scroll. This is by design, and is intentional. It creates a strong visual experience for the visitor when they first arrive.
If you don't like the look of this theme, it's likely easiest to choose a different theme - one that is better suited to the design you're looking for. For example, Dawn, Debut, and Supply all have a small header in their design.
If you prefer to stick with Brooklyn, but would like this section resized, hopefully another Community user will comment on the code necessary to make this change, as this is outside my scope of support. Otherwise, you can always hire a Shopify Expert for assistance. If you decide to stick with the Brooklyn theme, I'd recommend looking at the Brooklyn theme tips. These tips include image tips, which explain the best size for images in the theme for compatibility.
Let me know what you think of these suggestions!
Hilary | Community Moderator @ Shopify - Was my reply helpful? Click Like to let me know! - Was your question answered? Mark it as an Accepted Solution - To learn more visit the Shopify Help Center or the Shopify Blog