All the responses and not a one truly answered the question. Yes the alt tag is important to your site and SEO.
With the changes occurring in the HTML scripting language and the CSS styling languages over the last few years, the search engines, all of them now, are looking at page structural integrity and how the page is constructed as part of the SEO. HTML5 incorporated a large group of new tags based on recommendations by the IETF (Internet Engineering Task Force), those recommendations were based on many things, but the biggest being proper structural content.
Anyone that has been a web developer for any length of time knows, proper structuring of HTML content, that is strictly used for SEO. Laying out the page in such a way that the tags themselves identify much of the content. Not Div's classes and ID's but actual tags. That was the reason for turning the scripting language on its ear for a few years and incorporating all those new tags. It only makes sense if you think about it, think how much easier it is for the search engines to crawl a site if the tags properly identify the content with consistent names.
The attributes are a part of those tags and are very important to accessibility and responsiveness.
That includes your alt attributes, it is not a tag, it is an attribute, always has been, and a very important one at that. The alt attribute not only describes the image for the viewer in the event it cannot be downloaded but more importantly used for screen readers and accessibility for seeing disabled users of the web. That is the key reason for the alt attribute. Ignore it at your peril.
For anyone not paying attention, the two biggest areas of penalties at the current time, and we all know how fluid this is when it comes to SEO, on a website are mobile responsiveness and not following accessibility guidelines as set forth in the IETF.