The members of the German branch of the mastiff family have enjoyed steady popularity for a number of good reasons not least of which is their pleasing personalities. The younger of the two breeds, the Boxer, was developed in the 19th century; they are known for their affinity for people in general and children in particular.
The Boxer is an extremely energetic breed with a sense of humor; they enjoy playing and running. A young Boxer can be a boisterous dog to house around small children. The breed, however, is devoted to young people and have a natural protective instinct when it comes to watching over children.
The brindle Boxer I used to live with, Keeper, would watch over all children in her vicinity, regardless of who they belonged to. She once offered to protect a squealing child from the father who was tossing her in the air causing her to squeal. Yes, that protective. In other words, the Boxer needs firm and consistent handling so that their energy and instincts are channeled in constructive ways.
As devoted family dogs who need to be near their people, Boxers can develop separation anxiety if left home alone too long and too often. They benefit from several walks - or runs - a day, and enjoy opportunities to play every day too. My sister's family is living with their second adopted Boxer and every morning my niece gets up to take him jogging before school; once he's had his exercise for the morning he is able to nap until she comes home from school and takes him jogging again in the late afternoon.
He then needs an evening walk, plus he has become the neighborhood dog who plays with all the children after school. He cannot, however, be allowed off leash as he will run and not immediately return.
While once a tragically short lived breed, careful breeding has vastly improved the life expectancy of the breed - more and more Boxers are living to be 12 and 13 - for a breed that used to have an expectancy of 7 years this is a vast improvement.
Great Danes have been around longer than Boxers. The Dane is more directly related to ancient breeds but again, the Dane as we know them is a little more recent with probable influence from English Mastiff and Irish Wolfhound contributing to the modern breed.
This is another dog that does well with families, however, their size means they can easily knock over small children without meaning to. The Dane is less active than are Boxers but still require a walk and an opportunity to get out every day. Ironically, the Dane probably would adapt to a slightly smaller space more easily than the Boxer, as the Dane is much more prone to laying down on a couch and sleeping during the day, while a Boxer is far more active.
Either breed can become destructive if bored and both breeds are less likely to be bored if their people are around and interacting with them. These are both people centered breeds who do not do well if not given regular and consistent human interaction.
Both breeds also have a high incident of deafness and sometimes blindness in individuals who are white or predominately white. It is becoming commonplace to have hearing tests done on puppies who carry a white factor in their immediate genetic background.
Both breeds have regional breed rescues that can assist a family in finding the right family member to adopt. These breeds also can be found from time to time in shelters and with the economy continuing to drag, I've witnessed more people needing to re-home their dogs and trying to arrange for private adoption.
If one has breed experience then private adoption might be a viable way to go, otherwise, I strongly recommend working with breed rescue or knowledgeable shelter staff. While these are both handsome dogs, neither is a breed that is particularly suited to a family with little or no dog experience - these just aren't 'starter' dogs.