In the early years of this country, few people lived to old age, but for those that did, "old-age security" meant having children or property. Having family living nearby wasn't often an issue. In those days before railroads and automobiles, families were large and few children ventured far from home.
Before 1800, less than 5% of the U.S. population lived in cities. Everyone else resided in rural areas where extended families could live together easily and cheaply. Generally, people worked for themselves. As a 1937 Social Security pamphlet described it, "The home of a pioneer family was a little world in itself. Members of the family were their own farm and factory workers, butchers, bakers, and barbers; policemen and firemen; often their own doctors and nurses, and sometimes their own teachers as well." (Social Security, 1937) You didn't need cash to survive in that economy, and families were fairly self-sufficient.
Source: U.S. Census Bureau(See 1800-1990: Changes in Urban/Rural U.S. Population.)
Having a family was the key to survival. As later described in that Social Security pamphlet,
"A young man then could hardly afford not to marry. He needed a wife as a business partner, children as helpers. In early New England not only spinsters but bachelors were under a cloud. Bachelors, in fact, were regarded with suspicion. Usually they had to live where the court told them to. Single people had to attach themselves to a family to get a chance to work for their living. Both the children and the old people earned their place at the family table." (Social Security, 1937)
Children were expected to split their earnings with, or otherwise provide for, their parents. If a parent needed care, the children were expected to provide it. Elderly people in need of care who were childless but wealthy could hire whatever help they needed. Dependent elderly people who could not be cared for by their own families could be 'boarded out' with surrogate families, and the adult children paid for the cost of that care. Those elderly who were poor and childless, or whose children refused or were unable to care for them, ended up dependent on charity or public welfare.
Slaves, of course, had neither families nor property. Their families were often split up, so it would have been impossible for children to look after their own parents. The quality of their life in old age was completely dependent on the master they worked for.