How important is adding an alt attribute to an image?

1) search engine indexing
when search engine trying to index your page, the bot is unable to understand what information is presented on the visual. "Alt" tag is an attrib that allows search engine to read your visual in text mode.

in order to allow search engine to know more about your page and get a better indexing result, adding ALT will definitely improve your search result.

2) Accessibility
in order to allow more users to read the page, including those with accessibility difficulties, we add ALT to visual so that their special browsers and device can read out the ALT as description so that they can know more about what's written on the page

Alt tags won't affect whether or not your site gets indexed.

BUT they can have an impact on how well your site is indexed on some of the search engines. The search engines that crawl sites looking for text will definitely pick up the alt tags so placing relevant keywords in the alt tags is very helpful.

Making sure I added Alt tags was part of an overall strategy I used to optimize one of my sites and it certainly seemed to help as I now have an organic #5 listing for one of my top keywords on Google.

I think Alt tags are a small part of the picture but it's an easy part of the picture to use so I would say "go for it".

I think i would help with search engines to add alts to you images, but that isn't the reason for why they added alts to images. The reason is if someone visits you site that disables images then the alt still tells them what is there so that they can still navigate or read what was supposed to be there.
The biggest reason of all is when people that are blind go to your website the tools that they use will read and tell them what the images are on your website, IF you have them.
So to enable your website for all users you want alt, not for search engines but for usablility.