We have two bassets, and although they're fantastic dogs and I adore ours, I'm slightly wary of recommending them to anyone who hasn't had direct experience of them before, because they're very different from your average Lab or spaniel.
Bassets do their own thing, they have a houndy smell, and they're big dogs, despite being low to the ground. BUT, they're also sweet-natured, gentle with kids, laid-back, happy to walk for miles then snooze for hours, and genuinely lovely to have around the place. They're real 'family member' dogs, and have proper personalities.
You can't really train them like you can train a gundog or collie - they'll do something to please you if it suits them too, or if you're offering a food reward. Our 6 month old puppy still isn't quite house trained but he will bring back his toys in return for some kibble. (After HOURS of repetition from me.) Because they're scent hounds, they tend not to go after cats, but love sniffing the most disgusting stuff. The only time our older dog has bolted was when she scented hare - the traditional hunting prey - it took an hour to find her. Otherwise, she walks to heel without a lead and is totally placid.
Again, because they're pack hounds, bassets don't like being alone and do better if there's someone at home most of the time. They're not 'barky' dogs, but they do have a distinctive arrrooo - great for deterring any casual intruder who will imagine you've got the Hound of the Baskervilles inside.
As far as health goes, we haven't had any vet dramas with ours (touch wood), but you do have to watch their weight, keep an eye on bloat, and clean inside their ears. And be careful where you buy from - like all pedigrees, check the same dogs don't appear too frequently in the lines. Sadly, because they're so cute, they're frequently puppy-farmed, so sickly puppies from badly inbred litters often turn up in basset rescue, once they've outgrown their cuteness.
Make sure you meet fully grown dogs, instead of adorable puppies; there's a huge difference, literally. If you Google for your nearest Basset Rescue (there are quite a few round the country), the rescue people will tell you all the horror stories about fridge raiding and slobbering... while probably owning about eight or nine hounds themselves. I can't really imagine us having any other breed now, to be honest.
(Sorry for the essay!)