Is the basement cat-proof, and is it warm enough? I'd be concerned, given that it's winter and you live in Illinois. If not, is it possible for you to safely (for his sake and yours) move him back to the "safe room" (which, by the way, was the correct approach when bringing home a new cat).
None of us can say if he's eating and drinking enough because you did not mention how much food you are giving him. Maine Coons are large-sized cats so they'll need more than the average amount of food. Since he's very shy AND has suddenly been placed in a completely new and scary (for him) environment, it is likely that his food and water intake might be very small for the time being, until he begins to settle in. That being said, you have to make sure that he's at least eating something, and drinking some water if you aren't giving him canned food. Cats can develop significant health issues if they go too long (24-48 hours) without food.
As for how long that will take, it depends on the cat. Since he's shy it might taken him awhile. There are some basic things that you can do to help him get acclimated, both to his new home and his new owners:
● Get a Feliway plug-in diffuser. This is a product that, when plugged into an outlet (like a lot of air fresheners on the market today), releases pheromones into the immediate environment. The pheromones are absolutely undetectable by humans, but they serve to calm most cats down and, consequently, make them feel more safe and secure, which is high on the list of cat needs. Feliway doesnâ€™t work for every single cat, but its success rate seems to be pretty high. The Feliway spray is a good alternative to the diffuser, but people seem to get better results with the diffuser.
● Instead of or in addition to the Feliway, you can put a drop or two of Bachâ€™s Rescue Remedy into his water. This would also serve to relax him and make him less anxious about his new surroundings. But of course youâ€™d need to be sure that heâ€™s actually drinking water. You could also rub it on the back of his ears or add it to his canned food. But you might frighten him by attempting the former, and may make his food less appealing with the latter.
● Spend some time with him in the basement or the safe room. Get down onto his level because your size can be intimidating to him. Read a book out loud, or do some chores quietly. Even just sitting there is fine. You want him to get used to your voice and presence, and you want him to not view you as a threat. Do not look at him directly in the eyes, as cats tend to see this as threatening behavior. If your eyes meet, you can slowly blink while maintaining your gaze. Cats do the exact same thing with each other to connote friendly intentions. Ignoring him is actually better than trying to force yourself on him, which is a natural tendency for humans to do. If your schedule allows you to visit him at regular intervals, thatâ€™s even better. Cats love routine.
● Play with him. Your best bet is to use a wand / fishing rod type toy. This allows you to interact with him, helping to build a bond, but makes him feel safer because you remain at armâ€™s length. Da Bird is widely considered to be the greatest interactive cat toy ever made, and might be a good starting point.
Really, these suggestions are just the tip of the iceberg. Iâ€™m sure other members will post with additional ideas. Good luck and let us know how things are going.