Bassets are great rabbit dogs. My buddy has a beagle and a basset that are great rabbit dogs. The basset is the A-10 of bunny hunting - they are low and slow, brush busting bastards.
A good dog will have natural ability, but to get a real hunting partner, you will need to hone his natural talents. How old is he? While he is young, you should focus on obedience. Nothing worse than chasing a hound that will not “come”. Keep your lessons short and frequent. Keep it fun and focus only on the things that he does right. NOTHING BAD COMES FROM YOU.
Three 10 minute sessions a day are a lot more effective than one 30 minute session. Dogs are like kids and have short attention spans. Put him in play scenarios that teach a lesson and focus on obedience. For example, teach him to come by tossing a toy in a long hallway and then command “come” as he grabs the toy and starts to bring it back to you (he cannot go anywhere else but back to you).
Try letting him chase a bunny around the yard. This will really get his natural abilities fired up. If you do not have a fenced yard (or access to one) try dragging a dead rabbit through the grass in a big circle (wear rubber knee boots and run the string through a 4’ piece of pvc tube so that he is not tracking you) and then take the dog to the starting point (get excited and command “here, here, here”)and set him on the track.
Try to get tehm excited and then command “track” and let him go. Start with short drags and progressively increase the distance. Buy a couple bunnies and keep them in a trash bag in your freezer (if you have a chest freezer). Take them out and let them thaw the day before you are going to train.
I would also start getting him used to loud noises, as you do not want a gun-shy dog. Do not take him to a trap range or shoot a gun over a rabbit until they are used to loud noises in general. I condition my dogs to loud noises by banging a pot with a metal spoon and making a big ruckus before feeding them. After you have been banging on the pot at dinner time for awhile, use a starter pistol to shoot in the air after he has tracked the rabbit a good distance away from you.
Eventually, your dog will associate loud noises with something good. You will be able to pop a shotgun over his head and he will not even blink. Just take it slow - stupid hurts and the road back from gunshyness is long and difficult (if at all possible).
When he gets a bit older and is tracking the dead rabbit relatively reliably, I would suggest that you try and find someone that has a beagle or basset and see if they will let your dog (just one - not a pack) run with theirs. Be careful though, as putting a young dog in a group of high octane dogs can be a bit of a trying experinece for a young pup. However, it can be a lot of fun. Few thing better than hearing the howl of beagles and bassets through the timber as they take chase after a rabbit.
Not sure how familiar you are with rabbit hunting, but we usually kick a rabbit up and get the dogs on the track and then spread out and wait for the rabbit to circle back. You should not have any concern of shooting the dog, as the bunny is usually way ahead of the dogs.
Good luck. I am envious! Have fun and enjoy the experience. Few thing better than watching the dog that you have trained put it all together. Even better when people brag on your hound. PM me if you need some rabbit recipes.