If I want my site to be no.1 for a particular keyword search, would it harm me or help me if I put
alt="keyword" in all of the images I have in the site?
It might help you, though probably not nearly as much as you'd like to think. What's more important is that it would be abuse of what alt text is actually for: to serve as a replacement for the image if it doesn't load, and as an accessibility feature for visitors who may be using alternative(eg. screen reader) browsers.
In other words: don't do this.
It would help you. Alt tags are how search engines know what an image is about, and including them will help your site get ranked higher in image searches. If you don't include alt tags, your site will rank lower in image searches. If you want to get traffic from google images (and other image searches), then you should include alt tags.
On an somewhat unrelated note, disabled people can't see images, and having an alt tag would help them know what the image is about (they can read the alt tag). If people who can't see images are reading your site, then you should include alt tags to help them out.
But if you aren't worried about accessibility or image searches, then don't bother.