There might be a misunderstanding in some posts as to what alt (alternative) text is.
Bridge has a Metadata panel where you can enter information that can be embedded into some image file formats like JPEG. If you select an image and enter a Title, Description, Keywords, Copyright, etc. in the Metadata panel, when you upload the image to a website that knows how to extract that information (such as Flickr), the website may display the information with the photo on a web page.
But that's not alt text. Web browsers look for alt text in the HTML/CSS code that places an image on a web page, not from inside the image itself. Bridge doesn't edit web page code so technically, there is no way for Bridge to create alt text on its own.
Now, some websites may read the Description or Title in image metadata and convert it to alt text. If you're working with a web site that does that, and you know which field it converts, you could enter your text into the correct field in Bridge so that the website extracts it from the right place. But again, that alt text generation would be because of how that specific web site is coded to treat images; it would not be because entering alt text is possible in Bridge.
On the other hand, if your real goal is to be able to enter a description (caption), keywords, or other metadata that can be read and displayed by the web site you are working with, then Bridge can do that. But metadata inside an image is not the same as the text in the "alt" attribute in HTML code.
You can add 'metadata' to images, if that's what you mean. "Alt text" is an HTML/web thing.
You need to go into Bridge's menu: Edit > Preferences > Metadata, and enable IPTC Core -> Description.
Then you can add a Description in the Metadata panel. This is the industry standard way of describing images.