Great Dane or Staffy

The more I learn about purebreds and the more I work with our shelter dogs, the more I realize that there really is no guarantee for a "good with cats and kids" dog.  If your dog must be good with both, I'd recommend going to shelters and looking for a younger adult whose temperament and history are known.  I have three cats and plan to have kid eventually, so I had to give up on the idea of a puppy and instead found a 3 year old dog that has already lived 3 years around kids and cats, prefering their company.  Even still, I'm not yet convinced I can trust her with the cats and there will have to be a trial period where she proves that she won't snap.  I've made living arrangements to keep them seperated whenever they are not supervised if this becomes necessary.

Some breeds are generally more tolerant of children, other breeds generally have less prey drive, but I've seen Akitas snuggling babies and hound dogs snuggling cats so it all hinges on the invidual.

i think you kinda answered your own question about what to do. staffies are high energy for the most part.... they're terriers after all.
you didnt state how old your child is. some would say getting a puppy and letting it grow up with the kid is a great idea, and some say get a well seasoned bomb proof dog that has experience with kids... either way you go its still a gamble. dogs are allowed to change their mind too. i've seen some dogs that were great with kids to start with suddenly decide they have had enough. on the same token, i've seen dogs that hated kids finally give in and decide they arent so bad after all.
i grew up with both. my grand mother gave me and my brother a "middle aged" springer spanial named Fred. he had the personality of Fred from Smokey and the Bandit.... he tolerated all kinds of crap from us in his middle aged prime. but when my mom's sister moved in with us for a while, she brought her two daughters and her 2 yr old grand daughter. At this point Fred had hit his limit with kids crawling on him. he couldnt stand this kid at all. he never hurt her but he had the look in his eye that wouldnt mind it too much. my aunt had her own dog she brought with her. a little brindle pit bull named Katie. she was a spunky little dog, very high energy yet friendly. they claimed that was the 2 yr olds dog. only problem was this dog had way too much energy for a two year old. there were more than a few occasions when Katie would get so excited that she would run past the baby and knock her off her feet, which resulted in scared screaming baby. but her breed had nothing to do with that. any dog can knock a kid down in excitement. my sisters and i both took a mud bath in a local pond when our dad's Rhodesian Ridgeback rocketed past us and knocked us over. he was too busy having fun playing the water. he was NORMALLY a calm and gentle dog.... but he was very boisterous when he was happy.
Anyway, After the cousins moved out Fred died of old age a year or two later so we got a Shepherd mix, no idea what his dad was. but he was smart as a whip for a puppy. in one summer i had trained him to shake hands, sit stay lie down, play dead, but we hit hard times and had to find him a new home with a friend. a year later we needed a watch dog so we got a chow/husky/shepherd puppy. he was my pal. i was about 9 or 10 when we got him. he was a one person dog though. he couldnt care less about anyone else in the family but he shadowed my every move. he wasnt much for watching and protecting the home, but he made sure i was well looked after.

Yeah its tough though isnt it. The staffy would be brought up with the kid (shes nearly 7 so shes big enough and bright enough to be careful around the dog) and the cat and so should be ok with them, it is a breed that is known for tolerance to children, and is of course much smaller so cant do as much damage to them through innocent boisterousness. However the dane has, according to the last owners, proven it is good with kids and animals (its come from a farm with children) and is that bit older and more placid.
 
The watchdog thing is really a want rather than a must have, as long as he/she barks every now and again when disturbed Ill be happy and if it doesnt, it doesnt.
 
It basically comes down to a choice between a dog that could well be hyper and small, and a dog that is calm but would take up a quarter of our lounge. And its tough - we want a dog that is a manageable size, is calm, and is also available within our budget. Dont ask for much do we!

my sister has a great dane that is very good, very calm, placid, etc.
This dog will also eat your face if you try to get near my sister while she is sleeping. i know... i walked into the bedroom one night, not knowing she was asleep and Shirley growled quite fiercely at me. even if it was a bluff, most intruders wont push their luck.
and dont under estimate a staffy lol while at a dog show someone had a hyped up male staffy zig zagging his way through the crowd. i got permission to take his picture, he was very pretty, but when i crouched down to his level... he turned into a furry torpedo and knocked me on my rear!
after she pulled him off me, appologizing profusely(i didnt care.. he was licking, not biting) she got him to stand still with a treat so i
could get his picture. he was a ball of fire!!

If it's a Staffordshire (not an American staffordshire) they are known as 'nanny dogs' in england and other places, so that should just about answer your question if she'll be good with kids i also think the appearence of a bully dog works as a guard- and they are extrememly loyal dogs. I don't know much about danes but I know they have health issues and kinda short lifespans... but I really agree with everyone that its how you raise a dog, any dog can be good or bad with kids and cats.  my 2 pits love the cat. my female likes all kids but my male gets a little rough (just playin though) I think it all has to do with what they grow up with.