I always just take my bike with me on the bus. They can store them in the luggage compartment under the bus. Space is limited, so I probably shouldn't spread the word about it too much, but I've done it hundreds of times and there's only been one occasion (on a February school vacation week when there were about a dozen people taking their skis to the airport) where they didn't have space for my bike.
Logan Airport has two very good bike parking areas – links below point to the Google Street View of each location.
I should also note here that the Jetport has excellent covered bike parking, on the ground floor of the parking garage, in a high-visibility location right next to the terminal. I've ridden my bike there and left it for multiple days with no problems.
More generally, OpenStreetMap frequently will help you find bike parking. In the OpenCycleMap layer, bike racks are marked as blue dots, and sheltered bike racks are blue dots with a ^ over them. At close-in zoom levels it will also show the number of parking spaces available:
Christian MilNeil
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