Should a graduate with no experience offer to do menial work?

Doing menial work won't give you any more experience than doing group assignments in college, a summer internship, any extra curricular, etc.

If recent grads can't find an employer, the best thing for them to do would be to employ themselves. That is real experience, they'll find what they're good at & what they like doing. They might even find out that they don't need to find another employer. :)