Does civiCRM support RequiredIf validation for Profile fields
This was proposed as a feature for CiviCRM, but it never really got any funding: 'Other' option for custom fields

As Ben says, if you are (or have access to) a developer, it's not too hard to write this extension.  Add both the dropdown and the "Other" field to your profile, then use jQuery to hide the "Other" field.  Toggle whether it's hidden based on the value of the dropdown.

Use hook_civicrm_validateForm to set "Other" as required if the dropdown value matches the "Other" option.  By default it should not be required.