FamilySearch (https://www.familysearch.org/en/united-states/) has done an amazing job of making images of older Connecticut records accessible for free through the website. However, not everything will be accessible from home. When FamilySearch signs a contract to digitize records, they receive different levels of permission to share. Some of their contracts permit sharing from home, some from an affiliate organization, some from a Family History Center, some from the library in Salt Lake, and finally, some only to LDS members.
In general, Connecticut records are available either from home or from an affiliate. While there were will be some variation, these are the general categories:
Accessible from home: vital records, some probate, church records
Accessible from an affiliate: some probate, land records
To find an affiliate, go to https://locations.familysearch.org/en/search.
Connecticut Roots by Charter Oak Genealogy 