Can I modify my theme's header and footer with Beaver Builder?

It depends on which theme and which Beaver Builder products you have. 

Beaver Builder's Page Builder plugin is used for layouts in the content area of your site, not the theme areas (header and footer). See this knowledge base article for a diagram of the content area versus the parts of the page controlled by the WordPress theme. 

The Beaver Themer plugin is an add-on to the Page Builder plugin, and it can be used to create theme layouts to just about every area of the page besides the content area. You can create theme layouts for headers, footers, and other parts of the page, and you can control which pages in your site these layouts apply to. Here's an article with more information about the parts of the page that Beaver Themer can control. Beaver Themer overrides your theme's layout on the pages to which it applies, so your theme must support the removal of headers and footers with hooks. See the list of themes that are currently supported by Beaver Themer.

The Pro and Agency packages of Beaver Builder's Page Builder plugin include the Beaver Builder Theme, a very versatile theme framework that is simple to learn, the header and footer are very customizable if you don't use Beaver Themer, and it's fully compatible with Beaver Themer.

As an alternative to Beaver Themer, some Page Builder users like to suppress the header and footer in their theme and build their own header and footer using Page Builder modules, or they prefer to use a third-party header/footer plugin for Beaver Builder. Here's an article about creating a transparent header using Page Builder.