I understand how you feel but if you and your boyfriend are sharing a home I suggest you treat her the way you would if your daughter was sick. Get medicine, the vaporizer or humidifier which ever she needs, make sure she is eating food that will make her feel better and make sure she has lots of liquids. You could have her lay down until she is better, which lessens that chance of anybody else getting sick, of course you can make sure dad does it if you are worried about getting sick. Make sure to have plenty of antibacterial wipes to keep everything clean as possible.
No one who is sick should come around a newborn. It is just common sense for an adult to stay away when sick so a child should be no different. Just explain to him that when it happens if his daughter is sick she should not be around your newborn. I would tell my own mother the same thing. You do not want to risk a newborn getting a cold which turns into pneumonia. I wouldnt make a huge deal of it now though since when the time comes she could be healthy! But i would stand my ground if she is sick when your baby is born. Not worth the risk in my mind. Especially since kids dont understand how contagious things are and just cough and sneeze wherever.
I'd be taking her to the doctors and getting meds or give each her a better diet. It's not a good sign that the child is always sick but she also could have a very weak immune system. Up her vitamins I guess hope for your sake your baby will have a stronger immune system.
I would say that if I were in the situation (and I kind of am as far as having 3 stepdaughters), just start telling EVERYONE (your parents, your boyfriends parents, your boyfriend, friends and any other family members that may come around the baby) that anyone with a fever, runny nose or cough won't be allowed around your baby. If someone must be around the baby (your boyfriends daughter specifically) and they are sick, make them wear a mask and use lots and lots of soap and/or hand sanitizer. You can't be too cautious about a baby's health, especially a newborn. I wouldn't wait until the last minute to discuss it with your boyfriend either. If it's a subject that's likely to upset him, you're going to have to put you foot down and he's going to need time to accept it. You are the mother of the child and you have the right to protect it.
She should be treated as your child too. She shouldn't be kept from her father because shes sick!... all kids get sick if they're around other kids. Your precious baby isn't any better or more precious to him or shouldn't be anyway- get over yourself! Teach her not to touch the baby if shes sick and wash her hands... she sould be treated as if she is part of your family and as an older sibling.... because wether you like it or not.... she is.
I agree with Lanyb. She is part of your family too and its not her fault she gets sick. Try to teach good hygiene to help it from spreading to the baby, but its not bad if the baby gets a cold. It helps build their immune system.