As a web site owner you are responsible to secure your site, as all the other thing that you do in order to maintain your site you need to secure sites, this will ensure that users that use your site are safe from problematic malicious attacks and that your server is secure and not open to unwanted activities.
One of the easy method to secure sites is to keep all the related software up to date, this mean that if you are the owner of a blog that use word press as the site CMS (content management system), you need to install updated to your site, you need to verify that all the theme and the plugins are updated as most of the time new updated are fixing bug and security issues.
It is also good to use a secure socket layer, SSL certificate especially in the site login page, this way you give the end user a better security when he or she enter their username and password to your site. A SSL certificate, can be a shared one, this mean it is used by some not just you websites, this is the case in most of the share hosting in big hosting services and that what you need, till your site getting bigger and then you need to move on to your own dedicated/virtual server which include a SSL certificate with it, most of the time.
To most of the website owners, that use a ready to use CMS, like word press has a list of default plugins that they installed in each site, it is always good to add some security related plugins that will help keeping the site more secure.
There is another good to know free services that you can use to secure sites, there are free sites that can scan your site for malicious problem, vulnerabilities and even malwares that has been planted in your site, you can register your site on that websites and they will scan your site once a week, less or more, and send you a nice report about thing that they found, if there was something, also if your site allow user to upload files, it is very important to scan the files for malicious code before allowing other users use the files.
Keep in mind that there are tools that scan site all the times to check if the site is safe or not, so you, as the site owner, need to keep if safe, if one of the tools flag your site as a not safe site, it will be hard to you to make it safe again, not impossible but hard.