First things first. Google does not index certain file types such as CSS, JavaScript, images, and so on. But, you say, Google does index images. Well yes and no. It indexes images by using the text surrounding the image and the link values associated with the image. Yes, during growing pains while updating their capabilities, Google has indexed some of these files, but Google does not see it as content or directly valuable. This was discovered with many versions of the same JavaScript in it's index and the chatter began in earnest. I too found my CSS file indexed years ago, but it is long gone now. That is so 2009! It is old news that no longer applies. That is it. Nothing to worry about here.
Second, Google cannot access all Ajax elements- at least not Ajax enabled textbox(s) like what Google uses for search. In fact, they describe how to make Ajax elements crawlable here: https://developers.google.com/webmasters/ajax-crawling/. Google cannot and will not enter values into an Ajax enabled textbox searching for stuff. They actually say so. The scope of Google Ajax is well defined and began with Facebook and Disqus for comments. http://www.seroundtable.com/google-ajax-indexing-14241.html
Part of the confusion are early reports that Google executes JavaScript and can access Ajax. Where this is a true case is where JavaScript generates URLs. This was for three reasons; one to search deeper and not miss valid links, two to evaluate JavaScript for keyword loading, and three to check for JavaScript bloat. Later, other reasons were added such as page previews, and evaluating response times. It allows Google to index the full potential of a page. You can see the full scope here: https://www.distilled.net/blog/seo/google-stop-playing-the-jig-is-still-up-guest-post/
To be clear, so far Google is not abusing websites and you have nothing to worry about. It is not indexing elements that you would not want indexed. However, if you want Google and others, BTW- Bing does this too, from accessing these elements, then Google recommends putting in specific lines to each element/resource in your robots.txt file as normal. You can read the recommendation about half way down this page: http://googlewebmastercentral.blogspot.com/2011/11/get-post-and-safely-surfacing-more-of.html.
I would caution you against this however. Unless you are having a problem, I would not take any action. The reason answers the SEO part of your question. Blocking some elements may negatively impact Google and others from indexing your site properly and returning fully relevant results. If however, you have a JavaScript news element (real news and not site news), for example, that could in effect skew your search results. So in this case, you may want to restrict access to the JavaScript using robots.txt. This is where you have to put your thinking cap on. Take your time. There is no crisis to handle. When you have evaluated your site, then make informed decisions and take action. Mostly, you should not have to do anything at all unless a JavaScript result does not represent your content topic or Google and others are triggering a JavaScript and causing a problems.