Weak stems that cause perennials to fall over could be caused by one of several things. If they are not getting enough sunlight, the stems will stretch and be weak. This might be the case since you indicate that your Stella d'Oro daylilies don't rebloom. You may be able to correct excessive shade by pruning out branches of nearby trees and shrubs.
Another possibility is that the plants are getting too much nitrogen fertilizer. Even if you aren't applying a high-nitrogen fertilizer to the flower bed, it may be getting nitrogen fertilizer run off from the lawn. Avoid using fertilizers where the first number (the nitrogen content) is larger than the second (phosphorus) and third (potassium) numbers on the label.