When doing Basic Authentication, your browser will send an "Authorization" header. Its value is simply a Base64-encoded representation of "username:password" (like when you place credentials in the URL directly). Example for "user@example.com" with password "secret":

Authorization: Basic dXNlckBleGFtcGxlLmNvbTpzZWNyZXQ=

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So, in Ruby, you can create such headers like so:

Base64.strict_encode64("#{username}:#{password}")

Note that when speaking to a REST API, you should be using libraries like RestClient or HTTParty which will wrap Basic Auth and much more for you.