2 arseholes bassets here... if you have a good sense of humour, a strong stomach and the patience of a saint you might be alright! 
Eric (the short dumpy one) we've had from 10 weeks old. His finer moments include stealing my knickers/ bras/ socks/ tights and hiding them in a specific plantpot in the garden... i found his stash and retrieved them all so he retaliated by taking 1 of each of my pairs of shoes and hiding them back in his plant pot 
I've also attached a photo of what he looks like after a standard 30 minute walk.. This particular day was a heady bouquet or fox shit, rotting seagull corpse and canal water.. all 3 of which he's found and rolled in within 12 minutes of being outside the house.
Elvis (the taller one) is a rescue. When he came to us he was so scared of everything he'd wet himself if you spoke to him. Now he never bloody shuts up. Prime example: He decided that the buzzer noise on child genius (the TV show) was definately going to kill us all.. so spend the entire hour it was on barking in my ear every time the buzzer sounded.
Elvis also once ran into my living room with a banana. This would be fairly innocuous, were it not for the fact we dont buy or eat bananas and I was fairly certain there weren't any in the house. It turned out he'd escaped from the garden and stolen the banana from the boot of a neighbors car when she was taking her shopping inside. 
All joking aside, bassets are not for the faint hearted. They are clingy, completely and utterly untrainable for the most part due to a lack of common sense, and will drive you up the wall for the next 10 years. I managed to get mine faily well trained, but whereas you might need to do 100 repetitions of a command with a normal dog, you need to triple that for a basset.
Then there's the health issues. Eric is well bred because we were able to choose the breeder and research carefully, so he's a healthy chap. Elvis is another story. Elv as I mentioned is a rescue and therefore we didn't know his breeding before he landed in our lives. He's not quite 2 yet and has cost us about £2k in vet bills (thanks the lord for insurance!) thus far. He has short ulner syndrome, so his foot rotates outwards about 90 degrees which causes him pain if he's walked too far. He also has an autoimmune condition. Both could be prevented by better breeding so if you do go for a basset, research like mad and go for a well established, health tested line. Buckadoe bassets (Eric is a buckadoe) are excellent, as are wolferloe and many others, so find a line you like and trust me, if the puppy is on sale for less than £800, it's not well bred. PAY THE EXTRA, it will save you a fortune long term.
Other than that, bassets are dumpy, heart stealing fur kids who will simultaneously ruin your life and make you laugh until you hurt at the same time :)