Build the trip around daytime Key West
Walk or bicycle through Old Town in the morning, when porches, gardens, wooden houses, and wandering chickens are easier to notice. Explore the Historic Seaport before the day fills in, visit the cemetery or museums, tour Fort Zachary Taylor, follow literary history, or spend part of the day sailing, snorkeling, fishing, or paddling. None of these experiences requires alcohol.
Sunset is not limited to a bar
Mallory Square's Sunset Celebration is a public waterfront gathering with performers, vendors, and crowds. A sailing trip, waterfront walk, park visit, or quieter viewpoint can also make sunset part of the day without centering alcohol.
Location affects the nighttime experience
Staying directly beside the busiest blocks of Duval can expose you to late-night noise and crowds. A property elsewhere in Old Town can preserve walkability while reducing nightlife exposure. Verify the immediate block, not merely an "Old Town" label.
Do not avoid Duval completely
Duval contains galleries, shops, restaurants, architecture, and cultural landmarks as well as bars. Visit at the time of day that matches your interests, then step onto the parallel streets and waterfront. The point is not to avoid Key West's loud side, but to keep it from becoming the only Key West you see.