There are some clever stem supports you can buy at any of the big box stores that will hold up one heavy stem.
But I agree with Art, I think it's either watering not enough or too much. Looks like they're wet from rain? You can go out and gently! shake the blossoms after the rain stops, to rid them of the excess water that's weighing them down. If it keeps raining, you could spread some plastic around the base of the plant to help the soil dry out a bit. Don't forget to remove it when the weather dries out though.
Agapanthus prefer to be too dry, over too wet most of the time.