Basil will definitely not overwinter. It needs heat and sun and most species and cultivars are annuals. Rosemary is a woody evergreen perennial and will survive for years in a conducive climate. They don't 'come back' as such - they remain green all year round. But I don't know if it would survive in your climate. Do you know what cultivar you have?

Basil is an annual, so will not return next year. Rosemary is perennial, but does not like cold weather. You can try a heavy mulch.

Rosemary should survive a zone 7 winter without a lot of problems. The biggest concern would be wet soil........this plant really needs very good drainage. I would be concerned about a heavy mulch holding too much moisture against the root system so if you do mulch, make it a very coarse textured mulch like bark chunks or coarse wood chips.

Good snow cover will help.

Saturn, are you sure that you are in zone 7 in Rhode Island? Even if you're borderline zone 6, you might have luck with overwintering if you've got a hardy variety, and you probably will have to give it some protection as well as excellent drainage, as gardengal mentioned.

South coastal and eastern RI are in fact Zone 7.

The variety 'ARP' is known for its hardiness.

It should survive unless there is particularly wet and cold winter, mine have not had an issue with snow in and of itself. Drainage and soil is important as mentioned above.