Adding secure hosting (HTTPS) will not help with that. All it does is encrypt data traveling between your web site and your viewers — this is important for web sites that deal with confidential information, but if most of your viewers are just looking at your site, it isn’t particularly helpful.
Given what you’ve said, you’ll get much further by:
 Be sure that any software you’ve installed via One-Clicks is set to automatically update, and any web applications that you’ve installed yourself is kept up to date. Old, insecure versions of web apps are the #1 cause of hacked web sites.
 Set a secure password on your DreamHost Panel login, as well as on your FTP account. If you don’t use FTP to manage your site, set your user to SFTP-only.
 Make sure that you have antivirus software installed and working on your own computer — some viruses can steal passwords from your computer to deface your web site.
 Also make sure that your account and any domain registrations you have with us are set to renew automatically if appropriate. (We will email you when your account or domains come up for renewal, but if you miss that email you may be in for an unpleasant surprise.)