At my startup I have a partner and a few employees, all of whom have some amount of equity in the company. At what point (if ever) in my startup's life cycle should I stop offering (or significantly reduce) equity to new hires?
When you have enough people who will work for you for market wages and do not want options. Options are a reward for early in and extra incentive to join.
You should never have used options in a startup.
Options are only appropriate when there is an underlying value to the company. The value is set when the company is making money or raises Series A. After that, people should be getting a market wage and you can use an option program as a bonus based on their compensation program. Options are useful if you anticipate a future liquidation event.
Before Series A or breakeven you can use equity instead of paying people fair market rates. When someone contributes to your company and does not get paid, they are, in effect, "betting" on the future of the company. The value of their bet is equal to the unpaid portion of the fair market value of their contribution.
Their % equity should be based on their % of the bets.
This model, called Slicing Pie, is a much cleaner way of rewarding early employees. Option programs for startups will make your head explode!