My daughter often has a runny nose. It is disgusting! We try to keep it cleaned up, but until she learns to control the dripping and blow her own nose there are times when her face is just covered in snot. I suppose it looks to others like i don't care - the reality is having a two year old who doesn't want to spend life with a kleenex held to her face all day.
The other thing is, this runny nose is almost never accompanied by any sort of real illness. (Despite the very common runny nose, my daughter has been sick enough to complain or act off only a few days of her life. Other kids that she plays with do not seem to develop the same runny nose, so it does not appear that she is a typhoid-toddler. When other kids get an illness that make them quite sick, and she is exposed to them, she seems to be able to fight it off and not propagate the illness.)
Our doctor is of the opinion that it is the result of the bone structure in her
face that makes her very prone to getting runny noses, more than any underlying illness. Clearly any very light cold causes her to drip when another kid may not exhibit any symptoms externally. We have also considered that it may be the result of some allergies, for which we wouldn't pursue treatment at this age unless it bothered her more. The doctor does not recommend quarantining her when she has this symptom.
In short, i do understand where you are coming from. However, from my view point, my kid could have been that kid at story time, but i still feel like i can take her when she has a runny nose because a) it does not necessarily connote a contagious illness of any note, and b) we would be confined to the home 50% of the year if we didn't. I do my best to keep her clean, and i particularly have her practice good hygiene with hand washing and cough covering.
I know that's a lot of words to answer your question, but I hope that it is a little reassuring. I felt compelled to tell the story so you could see a side where having your toddler out with a runny nose wasn't the sin it might appear at first.
I don't think you need to be too concerned with a drippy clear snot, despite its appearances, because my experience suggests that it looks grosser than it actually is. Still, your best defense is to teach your child to practice good hygiene herself - washing hands every time you get home, before you eat, etc