This is an annoyance that most homeowners have to deal with from time to time as the washers in the faucets wear out with use. You can extend the life of your faucet washers by not turning them off with too much force.
You may also consider replacing your older faucets with those that feature washer less valve cartridges instead of rubber washers. Since the valve mechanism is contained in one cartridge, these faucets are easier to repair and replace. In addition, cartridges are generally more durable and do not leak because they do not deteriorate with friction or age as washers do.