Would enabling 'Use Ajax' for view content harm our SEO?
It will only have an effect if there is content that you want to be visible to the search engine crawlers that is only accessible after having run any JavaScript (aka an ajax call).

For example maybe you want to be optimising for the key word carrot, and there is lots of content about carrots that only appears on page 2 of your view. To get to page two you have to do an ajax request so crawlers will never get there and it will in this instance negatively impact your SEO.

This is quite specific though, and could easily be mitigated against by ensuring your important content on carrots appears on the first page of the view.

So I guess it could have an impact, but it's unlikely that it will be a very big one. But certainly something to consider.

Google has a tool that allows you to fetch a page of your website and preview how it looks to the crawler. 
This could be useful for testing exactly what impact enabling ajax on your view will have.