This issue seems to only happen on private instances. (E.g: Github Enterprise, Stash)
If the password contains an equals sign, '=', access over HTTP will always prompt to enter password on Sourcetree, sometimes twice.
Using the Keychain Access app, there's an entry under the private instance server name. However, the stored password is truncated where the equal sign should be. Manually entering the full password there in Keychain Access works around the issue.