How to care of umbilical cord?

Okay so I just had my baby four days ago, thing were going fine but now I don't know if I'm overreacting to his umbilical cord. So I'm cleaning with alcohol wipes like the doctor told me, every time I change his diaper. My concern is that well they on the clamp ( not sure that's the name) and it seems like it's pulling it down. I'm worried that it's not suppose to do that, I am overreacting or is it okay? Also, how did you take care of your little ones cords?