In the early 1960's, when Betty and Peter's relationship took place, it was not uncommon for young men and women to take on full time jobs. People in their late teens were generally expected to grow up more quickly and assume more responsibility than following generations have. In the series "Untold Tales of Spider-Man", it is explained that Betty is indeed Peter's age, but had since dropped out of school.
As far as explaining how Betty could be Peter's age or only slightly older in a more contemporary setting...I don't think it would require a huge suspension of disbelief to understand that Betty could have graduated high school but opted not to immediately continue on to college (for whatever reason, to take care of her sick mother, the story line with her brother, Bennet, etc...) and instead take on secretarial duties at the Bugle (which definitely don't require a higher education). Heck, maybe in this contemporary setting, Betty is even taking college classes at night.