Cookies are little bits of information that websites store on your computer when you visit them. A website might use a cookie to store the fact that you've logged in to their website; that way when you return, you don't need to log in again. If it's an e-commerce website, it might store your currency preference. Cookies only contain information that you or the website itself provides. Cookies don't allow websites any extra access to your files or personal information other than what you provide it with.
Remember; your cookies setting is a "per browser" setting; so you've got them enabled in Webkit based browser, but they might be still disabled in a different web browser on your computer (...if you've got more than one web browser installed!)
What are my cookie settings?
What is the point of using cookies on a website?
What's with all those annoying messages?
Block some or all cookies by disabling them
Are other websites able to read my cookies?
Is there a need to be worried about cookies?
Cookies are less of a "safety" issue, and more of a "potential for tracking you/privacy" issue.
For the most part, cookies are a safe and necessary part of using the Internet. A lot of websites won't work properly if you don't have Cookies enabled.
Cookies (particularly Third-Party Cookies) can be used to track your behaviour between websites (such as in the example of advertisers building a profile of your activity over time).
If you are concerned about Cookie safety, you can disable cookies and try browsing the web. Notice if you have different experiences with websites that you visit. If you're unable to access certain websites because of this, then consider turning them back on. If not, then feel free to leave cookies disabled!
If you are interested in having a lot of control about which websites place cookies on your computer, you can install an extension in some web browsers (like Chrome or Firefox) to help you manage them.
Generally speaking, I have cookies enabled in my web browser and I advise my clients, friends and familes to keep the enabled as well.