Well the question would be the integrity of the website really, not necessarily the connection. But if you want to make sure you're using a secure connection to a website, you have the ability to enable a type of security precaution that encrypts all the data that's sent and received from your computer to the website and back. This is usually represented by an "https" instead of "http", with the 's' standing for secure, from within the website address.
More times than not the setting is already enabled within your browser, so it comes down to if the site supports that kind of security feature. Go to paypal.com and you'll see it, whereas on espn.com you won't. Usually sites that hold sensitive information about the user, like facebook, or those that host merchant services, tend to use https as a security measure. If you want to see a little information about your secure connection, you can usually click on the little padlock just next to the site address.